What does the word combination mean

Princeton’s WordNetRate this definition:4.2 / 6 votes

  1. combinationnoun

    a collection of things that have been combined; an assemblage of separate parts or qualities

  2. combinationnoun

    a coordinated sequence of chess moves

  3. combinationnoun

    a sequence of numbers or letters that opens a combination lock

    «he forgot the combination to the safe»

  4. combinationnoun

    a group of people (often temporary) having a common purpose

    «they were a winning combination»

  5. combinationnoun

    an alliance of people or corporations or countries for a special purpose (formerly to achieve some antisocial end but now for general political or economic purposes)

  6. combinationnoun

    the act of arranging elements into specified groups without regard to order

  7. combination, combining, compoundingnoun

    the act of combining things to form a new whole

WiktionaryRate this definition:0.0 / 0 votes

  1. combinationnoun

    The act of combining, the state of being combined or the result of combining.

  2. combinationnoun

    A sequence of numbers or letters used to open a combination lock.

  3. combinationnoun

    One or more elements selected from a set without regard to the order of selection.

  4. combinationnoun

    An association or alliance of people for some common purpose.

  5. combinationnoun

    A combination shot; a billiard; a shot where the cue ball hits a ball that strikes another ball on the table.

Samuel Johnson’s DictionaryRate this definition:0.0 / 0 votes

  1. Combinationnoun

    Etymology: from combine.

    1. Union for some certain purpose; association; league. A combination is of private persons, a confederacy of states or sovereigns.

    This cunning cardinal
    The articles o’ th’ combination drew,
    As himself pleas’d.
    William Shakespeare, Henry VIII.

    2. It is now generally used in an ill sense; but was formerly indifferent.

    They aim to subdue all to their own will and power, under the disguises of holy combinations.
    Charles I .

    3. Union of bodies, or qualities; commixture; conjunction.

    These natures, from the moment of their first combination, have been and are for ever inseparable.
    Richard Hooker, b. v. s. 52.

    Resolution of compound bodies by fire, does not so much enrich mankind as it divides the bodies; as upon the score of its making new compounds by new combinations.
    Boyle.

    Ingratitude is always in combination with pride and hard-heartedness.
    Robert South, Sermons.

    4. Copulation of ideas in the mind.

    They never suffer any ideas to be joined in their understandings, in any other or stronger combination than what their own nature and correspondence give them.
    John Locke.

    5.Combination is used in mathematicks, to denote the variation or alteration of any number of quantities, letters, sounds, or the like, in all the different manners possible. thus the number of possible changes or combinations of the twenty-four letters of the alphabet, taken first two by two, then three by three, &c. amount to 1,391,724,288,887,252,999,425,128,493,402,200. Ephraim Chambers

Webster DictionaryRate this definition:1.0 / 1 vote

  1. Combinationnoun

    the act or process of combining or uniting persons and things

  2. Combinationnoun

    the result of combining or uniting; union of persons or things; esp. a union or alliance of persons or states to effect some purpose; — usually in a bad sense

  3. Combinationnoun

    the act or process of uniting by chemical affinity, by which substances unite with each other in definite proportions by weight to form distinct compounds

  4. Combinationnoun

    the different arrangements of a number of objects, as letters, into groups

FreebaseRate this definition:0.0 / 0 votes

  1. Combination

    In mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruit, say an apple, orange and pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
    which can be written using factorials as whenever, and which is zero when . The set of all k-combinations of a set S is sometimes denoted by .
    Combinations can refer to the combination of n things taken k at a time without or with repetitions. In the above example repetitions were not allowed. If however it was possible to have two of any one kind of fruit there would be 3 more combinations: one with two apples, one with two oranges, and one with two pears.
    With large sets, it becomes necessary to use more sophisticated mathematics to find the number of combinations. For example, a poker hand can be described as a 5-combination of cards from a 52 card deck. The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.

Editors ContributionRate this definition:0.0 / 0 votes

  1. combination

    To combine into a whole.

    There are a perfect combination of personalities and it is such joyful news to know they are getting married.

    Submitted by MaryC on March 17, 2020  

Matched Categories

    • Alliance
    • Change Of Integrity
    • Collection
    • Mathematical Process
    • Politics
    • Sequence
    • Unit

British National Corpus

  1. Spoken Corpus Frequency

    Rank popularity for the word ‘combination’ in Spoken Corpus Frequency: #2275

  2. Written Corpus Frequency

    Rank popularity for the word ‘combination’ in Written Corpus Frequency: #4319

  3. Nouns Frequency

    Rank popularity for the word ‘combination’ in Nouns Frequency: #861

How to pronounce combination?

How to say combination in sign language?

Numerology

  1. Chaldean Numerology

    The numerical value of combination in Chaldean Numerology is: 4

  2. Pythagorean Numerology

    The numerical value of combination in Pythagorean Numerology is: 7

Examples of combination in a Sentence

  1. Conductor Ludwig Wicki:

    For me it’s a combination of music and horses and landscapes. When I was young I dreamed to be a cowboy, and I think that’s what Western is, it’s a dream for mostly boys, to be a cowboy or an Indian, and have these beautiful clothes, we played that in the forest, and that’s still in my heart.

  2. Gwen Stefani:

    I wonder … it’s the combination. It’s the chemistry.

  3. Rick Molinari:

    You might think you’re tipsy, but it’s the combination of the sugar and the caffeine.

  4. Nikita Shah:

    Patients willsay I started taking tamoxifen and I gained 40 pounds, but that’s probably a combination of hormonal and lifestyle changes they’re going through. Tamoxifen can cause some fluid retention, and it could maybe cause a 2- or 3-pound weight gain, but beyond that there are probably other things going on as well.

  5. Kate Larsen:

    In an ideal world you use both to have the most efficient outcome, some combination of those two approaches is really going to be what’s most effective at deploying clean generation at the scale we need.

Popularity rank by frequency of use


Translations for combination

From our Multilingual Translation Dictionary

  • مزيجArabic
  • съединение, комбинация, обединениеBulgarian
  • combinacióCatalan, Valencian
  • kombinaceCzech
  • Kombination, VereinigungGerman
  • τρίκυκλοGreek
  • combinaciónSpanish
  • sobitama, järjestus, kombinatsioon, kooslus, kombineerimaEstonian
  • ترکیبPersian
  • ratsastus, kombinaatio, yhdistäminen, yhdistely, yhdistelmä, yhdistysFinnish
  • combinaisonFrench
  • combinaciónGalician
  • संयोगHindi
  • combinazioneItalian
  • 統合, 結合, 合併Japanese
  • combinatioLatin
  • kombinasjonNorwegian
  • vereniging, verbinding, combinatieDutch
  • kombinasjonNorwegian Nynorsk
  • combinaçãoPortuguese
  • combinație, combinareRomanian
  • сочетание, объединение, соединение, комбинацияRussian
  • kombinering, kombinationSwedish
  • birlik, dernek, birleşim, cemiyet, kombinasyon, birleştirmeTurkish

Get even more translations for combination »

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Are we missing a good definition for combination? Don’t keep it to yourself…

What word combination means?

noun. the act of combining or the state of being combined. a number of things combined: a combination of ideas. something formed by combining: A chord is a combination of notes. an alliance of persons or parties: a combination in restraint of trade.

What is an example of combination?

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination.

What kind of word is combination?

The act of combining, the state of being combined or the result of combining.

How can I use combination in a sentence?

A combination of factors led to her decision. The milkshakes come in chocolate, strawberry, and vanilla, or any combination of these. He has the right combination of talent and experience. The combination of these two chemicals can cause an explosion.

What’s another word for combo?

What is another word for combo?

stew blend
mixture meld
synthesis compound
composite alloy
combination admixture

What means perfect combination?

1 having all essential elements. 2 unblemished; faultless. a perfect gemstone. 3 correct or precise. perfect timing.

What is the opposite of combination?

Opposite of a joining or merging of different parts or qualities in which the component elements are individually distinct. detachment. permutation. separation. severance.

How do you combine words?

Does combination matter?

When the order doesn’t matter, it is a Combination. When the order does matter it is a Permutation.

What is the difference between combination and permutation?

A permutation is an act of arranging the objects or numbers in order. Combinations are the way of selecting the objects or numbers from a group of objects or collection, in such a way that the order of the objects does not matter.

What is the meaning of permutation and combination?

permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.

How do combinations work?

How do you solve combination?

What is the importance of combination?

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order. Combinations can be confused with permutations.

How do you do 10 Pick 3?

How many combinations are there in math?

The formula for combinations is generally n! / (r! (n — r)!), where n is the total number of possibilities to start and r is the number of selections made. In our example, we have 52 cards; therefore, n = 52.

How many combinations of 8 numbers are there?

Note: 8 items have a total of 40,320 different combinations. For the sake of output and server capacity, we cannot let you enter more than 8 items!

How many combinations of 3 numbers are there?

There are, you see, 3 x 2 x 1 = 6 possible ways of arranging the three digits. Therefore in that set of 720 possibilities, each unique combination of three digits is represented 6 times. So we just divide by 6. 720 / 6 = 120.

How many combinations can you make with 3 numbers without repeating?

If you’re excluding 3-digit numbers with leading zeros, then you can have 900 combinations of 3 numbers without repetition. The numbers 100 through 999. If you’re not excluding 3-digit numbers with leading zeros, then you can have 1,000 combinations of 3 numbers without repetition.

What does 5 choose 3 mean?

5C3 or 5 choose 3 refers to how many combinations are possible from 5 items, taken 3 at a time.

How do you figure out how many combinations a lock has?

How many combinations are there with 10 digits?

If repetition is allowed, then the number of permutations of 10 digits is 10,000,000,000. If repetition is not allowed, then the number of permutations of 10 digits is 3,628,800.

Last Update: Jan 03, 2023

This is a question our experts keep getting from time to time. Now, we have got the complete detailed explanation and answer for everyone, who is interested!


Asked by: Brennon Lang

Score: 4.8/5
(71 votes)

In mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter.

What does any combination mean?

a collection of things that have been combined; an assemblage of separate parts or qualities. combinationnoun.

What is word combination?

Many scientific words are based either on combinations of prefixes/suffixes of Greek, Latin, Indo-European or other origin linked to descriptive words or are combinations of short independent words included together as part of descriptive phrases.

What is combination with example?

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. … Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC.

What does combinations mean in math?

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order.

23 related questions found

How does combination work?

Combinations are a way to calculate the total outcomes of an event where order of the outcomes does not matter. To calculate combinations, we will use the formula nCr = n! / r! * (n — r)!, where n represents the total number of items, and r represents the number of items being chosen at a time.

How do you represent a combination?

These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2n − 1, where each digit position is an item from the set of n. Representing these subsets (in the same order) as base 2 numerals: 0 – 000. 1 – 001.

What is the example of combination reaction?

When a combination reaction occurs between a metal and a non-metal the product is an ionic solid. An example could be lithium reacting with sulfur to give lithium sulfide. When magnesium burns in air, the atoms of the metal combine with the gas oxygen to produce magnesium oxide.

What are the types of combination?

Combination reactions are of three types:

  • In the first type of combination reaction, an element combines with another element to form a new compound. …
  • In second type of combination reaction, an element combines with a compound. …
  • In the third type of combination, 2 or more compounds combine to form a new compound.

What is a combination problem?

Stated generally, the combination problem is the problem of how precisely the fundamental conscious minds come to compose, constitute, or give rise to some further, additional conscious mind (especially our own).

What is it called when 2 words are combined?

Portmanteau word, also called blend, a word that results from blending two or more words, or parts of words, such that the portmanteau word expresses some combination of the meaning of its parts.

What is it called when you put two words together to make one?

listen), /ˌpɔːrtmænˈtoʊ/) or portmanteau word (from «portmanteau (luggage)») is a blend of words in which parts of multiple words are combined into a new word, as in smog, coined by blending smoke and fog, or motel, from motor and hotel. …

What is perfect combination?

1 having all essential elements. 2 unblemished; faultless.

Can you use combination in a sentence?

A combination of factors led to her decision. The milkshakes come in chocolate, strawberry, and vanilla, or any combination of these. He has the right combination of talent and experience. The combination of these two chemicals can cause an explosion.

What is combination or combinational mean?

Definitions of combinational. adjective. able to or tending to combine. synonyms: combinable, combinatory combinative, combinatory. marked by or relating to or resulting from combination.

What is a combination in statistics?

A combination is a selection of all or part of a set of objects, without regard to the order in which they were selected. This means that XYZ is considered the same combination as ZYX.

Which is the combination reaction?

A combination reaction is a reaction in which two reactants combine to form one product. … When a metal undergos a combination reaction with oxygen, a metal oxide is formed (similarily, a metal halide is formed if reacted with one of the halogens).

What is combination reaction and types?

Combination Reaction

A reaction in which two or more reactants combine to form a single product is known as a combination reaction. It takes the form of X + Y → XY. Combination reaction is also known as a synthesis reaction. Example of combination reaction: 2Na + Cl2 → 2NaCl.

Which is not a combination reaction?

The second reaction is not an example of combination reaction. (AgNO3+NaCl —>AgCl + NaNO3.) … But,in the second reaction,only the NO3 and Cl are interchanging their places to make the chemical products of the reaction. So,here no combination of the reactants is happening.

What is direct combination?

A synthesis reaction or direct combination reaction is a type of chemical reaction in which two or more simple substances combine to form a more complex product. The reactants may be elements or compounds, while the product is always a compound.

What are combination reaction explain with an equation?

A combination reaction (also known as a synthesis reaction) is a reaction where two or more elements or compounds (reactants) combine to form a single compound (product). Such reactions are represented by equations of the following form: X + Y → XY. … Combination reaction is also known as a synthesis reaction.

What is combination reaction give two examples?

In combination reactions, two or more substances combine to form a new compound. Only one product is obtained in such reactions. The reactants in such reactions can be elements as well as compounds. For Example, Calcium oxide reacts with water to produce calcium hydroxide.

Does combination matter?

When the order doesn’t matter, it is a Combination. When the order does matter it is a Permutation.

Is repetition allowed in combination?

Permutations: order matters, repetitions are not allowed. (regular) Combinations: order does NOT matter, repetitions are not allowed. Combinations WITH Repetitions: order does NOT matter, repetitions ARE allowed.

комбинация, сочетание, соединение, объединение, совмещение, союз

существительное

- сочетание, комбинация

in combination — в сочетании, во взаимодействии
phraseological combination — лингв. фразеологическое сочетание

- союз, объединение, общество
- клика, шайка, банда
- обыкн. pl женское нижнее бельё, трико, комбинация (типа цельного купальника) (тж. combination garment)
- комбинированный прибор, комбинированное устройство и т. п.

combination body — авт. кузов со съёмным верхом

- мотоцикл с прицепной коляской
- цифровая или буквенная комбинация (замка)

the combination of the safe — код замка (этого) сейфа

- замок с секретом, секретный замок
- мат. сочетание
- хим. соединение

combination reaction — реакция присоединения

- разг. см. combination room

Мои примеры

Словосочетания

a combination of intrinsic and extrinsic factors — сочетание внутренних и внешних факторов  
a heady combination of wine and brandy — пьянящее сочетание вина и коньяка  
combination tanned leather — кожа комбинированного дубления  
combination room — профессорская  
business combination — объединение компаний, слияние компаний  
combination block — комбинированное клише (растровое и штриховое)  
forbidden-combination check — контроль по запрещенным комбинациям  
cipher combination — комбинация шифра  
ciphertext combination — комбинация (группа знаков) шифротекста  
code combination — кодовая комбинация  
bit combination — набор двоичных знаков  
a combination of both — и тот, и другой  

Примеры с переводом

He forgot the combination to the safe.

Он забыл комбинацию к сейфу.

Blue and white is the classic colour combination and will never date.

Голубой и белый — это классическое сочетание цветов, которое никогда не выйдет из моды.

A combination of factors led to her decision.

К её решению привело совпадение нескольких факторов.

What is the combination to the safe?

Какая комбинация к сейфу?

The combination of warmth and music sent him to sleep.

Сочетание тепла и музыки усыпило его.

The combination to the safe was a secret.

Комбинация к сейфу хранилась в секрете.

He has the right combination of talent and experience.

В нём отлично сочетаются талант и опыт.

ещё 11 примеров свернуть

Примеры, ожидающие перевода

Chocolate and mint is a winning combination.

The colour combination just doesn’t work for me.

Reading is taught using a combination of several methods.

Для того чтобы добавить вариант перевода, кликните по иконке , напротив примера.

Возможные однокоренные слова

Формы слова

noun
ед. ч.(singular): combination
мн. ч.(plural): combinations

Other forms: combinations

A combination is a grouping together of separate things. Your desire to live to a big city might come from a combination of your dreams of being on stage, and your obsession with underground trains.

Combination is the act of combining, which comes from the Latin for «joining together two by two,» although it’s not necessary that you combine things in pairs. Your famous party mix might be a combination of pretzels, nuts, and espresso beans. Your locker combination is the set of numbers you have to dial to open the lock. If you plan to go on a TV talent show, get ready to learn some dance combinations. Kick, spin, drop, roll!

Definitions of combination

  1. noun

    the act of combining things to form a new whole

    synonyms:

    combining, compounding

    see moresee less

    types:

    show 22 types…
    hide 22 types…
    affixation, attachment

    the act of attaching or affixing something

    confusion

    an act causing a disorderly combination of elements with identities lost and distinctions blended

    admixture, commixture, intermixture, mix, mixing, mixture

    the act of mixing together

    fusion

    the act of fusing (or melting) together

    blend, blending

    the act of blending components together thoroughly

    interspersal, interspersion

    the act of combining one thing at intervals among other things

    temperance

    the act of tempering

    conjugation, jointure, unification, union, uniting

    the act of making or becoming a single unit

    consolidation, integration

    the act of combining into an integral whole

    graft, grafting

    the act of grafting something onto something else

    babel

    a confusion of voices and other sounds

    confluence, conflux, merging

    a flowing together

    homogenisation, homogenization

    the act of making something homogeneous or uniform in composition

    coalescence, coalescency, coalition, concretion, conglutination

    the union of diverse things into one body or form or group; the growing together of parts

    reunification, reunion

    the act of coming together again

    tribalisation, tribalization

    the act of making tribal; unification on a tribal basis

    umbrella

    having the function of uniting a group of similar things

    centralisation, centralization

    the act of consolidating power under a central control

    incorporation

    consolidating two or more things; union in (or into) one body

    amalgamation, merger, uniting

    the combination of two or more commercial companies

    vertical combination, vertical integration

    absorption into a single firm of several firms involved in all aspects of a product’s manufacture from raw materials to distribution

    horizontal combination, horizontal integration

    absorption into a single firm of several firms involved in the same level of production and sharing resources at that level

    type of:

    change of integrity

    the act of changing the unity or wholeness of something

  2. noun

    a collection of things that have been combined; an assemblage of separate parts or qualities

  3. noun

    the act of arranging elements into specified groups without regard to order

  4. noun

    a group of people (often temporary) having a common purpose

    “they were a winning
    combination

  5. noun

    an alliance of people or corporations or countries for a special purpose (formerly to achieve some antisocial end but now for general political or economic purposes)

  6. noun

    a sequence of numbers or letters that opens a combination lock

    “he forgot the
    combination to the safe”

  7. noun

    a coordinated sequence of chess moves

DISCLAIMER: These example sentences appear in various news sources and books to reflect the usage of the word ‘combination’.
Views expressed in the examples do not represent the opinion of Vocabulary.com or its editors.
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I’m an incurable romantic, and Casablanca’s one of the most romantic pictures I’ve ever seen — the combination of Bogart and Bergman is just magical.

Ken Adam

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PRONUNCIATION OF COMBINATION

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GRAMMATICAL CATEGORY OF COMBINATION

Combination is a noun.

A noun is a type of word the meaning of which determines reality. Nouns provide the names for all things: people, objects, sensations, feelings, etc.

WHAT DOES COMBINATION MEAN IN ENGLISH?

Combination

In mathematics, a combination is a way of selecting members from a grouping, such that the order of selection does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a kcombination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient which can be written using factorials as whenever, and which is zero when. The set of all k-combinations of a set S is sometimes denoted by. Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection, k-multiset, or k-combination with repetition are often used.


Definition of combination in the English dictionary

The first definition of combination in the dictionary is the act of combining or state of being combined. Other definition of combination is a union of separate parts, qualities, etc. Combination is also an alliance of people or parties; group having a common purpose.

WORDS THAT RHYME WITH COMBINATION

Synonyms and antonyms of combination in the English dictionary of synonyms

SYNONYMS OF «COMBINATION»

The following words have a similar or identical meaning as «combination» and belong to the same grammatical category.

Translation of «combination» into 25 languages

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TRANSLATION OF COMBINATION

Find out the translation of combination to 25 languages with our English multilingual translator.

The translations of combination from English to other languages presented in this section have been obtained through automatic statistical translation; where the essential translation unit is the word «combination» in English.

Translator English — Chinese


组合

1,325 millions of speakers

Translator English — Spanish


combinación

570 millions of speakers

Translator English — Hindi


संयोजन

380 millions of speakers

Translator English — Arabic


مَجْمُوعَة

280 millions of speakers

Translator English — Russian


сочетание

278 millions of speakers

Translator English — Portuguese


combinação

270 millions of speakers

Translator English — Bengali


combinate

260 millions of speakers

Translator English — French


combinaison

220 millions of speakers

Translator English — Malay


Menggabungkan

190 millions of speakers

Translator English — German


Kombination

180 millions of speakers

Translator English — Japanese


組み合わせ

130 millions of speakers

Translator English — Korean


결합

85 millions of speakers

Translator English — Javanese


Gabungke

85 millions of speakers

Translator English — Vietnamese


sự kết hợp

80 millions of speakers

Translator English — Tamil


கொண்டு combinate

75 millions of speakers

Translator English — Marathi


एकत्र करा

75 millions of speakers

Translator English — Turkish


combinate

70 millions of speakers

Translator English — Italian


combinazione

65 millions of speakers

Translator English — Polish


połączenie

50 millions of speakers

Translator English — Ukrainian


поєднання

40 millions of speakers

Translator English — Romanian


combinație

30 millions of speakers

Translator English — Greek


συνδυασμός

15 millions of speakers

Translator English — Afrikaans


kombinasie

14 millions of speakers

Translator English — Swedish


kombination

10 millions of speakers

Translator English — Norwegian


kombinasjon

5 millions of speakers

Trends of use of combination

TENDENCIES OF USE OF THE TERM «COMBINATION»

The term «combination» is very widely used and occupies the 3.971 position in our list of most widely used terms in the English dictionary.

Trends

FREQUENCY

Very widely used

The map shown above gives the frequency of use of the term «combination» in the different countries.

Principal search tendencies and common uses of combination

List of principal searches undertaken by users to access our English online dictionary and most widely used expressions with the word «combination».

FREQUENCY OF USE OF THE TERM «COMBINATION» OVER TIME

The graph expresses the annual evolution of the frequency of use of the word «combination» during the past 500 years. Its implementation is based on analysing how often the term «combination» appears in digitalised printed sources in English between the year 1500 and the present day.

Examples of use in the English literature, quotes and news about combination

10 QUOTES WITH «COMBINATION»

Famous quotes and sentences with the word combination.

I’m an incurable romantic, and Casablanca’s one of the most romantic pictures I’ve ever seen — the combination of Bogart and Bergman is just magical.

It’s such a strange combination that I’d be unhappy to make anything like that without Landis directing.

I don’t really drink sodas, but when I have popcorn or pizza I need a little. It’s the perfect combination.

I’m really not that weird. I’m a combination of a lot of different things. Maybe it’s just easier to make me look weird than another model who is specifically Caucasian.

I think only of objects: of a leg or an arm, of the wonderful sense of foreshortening, breaking through the plane, of the division of space, of the combination of straight lines in relation to curved ones.

The world is being re-shaped by the convergence of social, mobile, cloud, big data, community and other powerful forces. The combination of these technologies unlocks an incredible opportunity to connect everything together in a new way and is dramatically transforming the way we live and work.

It’s a combination of melody and lyrics, not one without the other. It’s a confluence of these different elements that makes something powerful.

I’m also interested in the modern suggestion that you can have a combination of love and sex in a marriage — which no previous society has ever believed.

As far as sleeping goes, you’re up and ready to go at six in the morning. Spring training was always a combination of relaxing and working, and I missed that quite a bit. I missed being around the ball field. A baseball. A bat. The smell of the uniform, you might say. Talking baseball. Seeing opponents as well as the Cubs.

A passion for politics stems usually from an insatiable need, either for power, or for friendship and adulation, or a combination of both.

10 ENGLISH BOOKS RELATING TO «COMBINATION»

Discover the use of combination in the following bibliographical selection. Books relating to combination and brief extracts from same to provide context of its use in English literature.

1

The Combination of Stellar Influences

The Combination of Stellar Influences is a well-known and popular book that explains the development of cosmobiology, which uses the 90-degree dial, and teaches the methods of this progressive school of astrology.

2

Combination Products: Regulatory Challenges and Successful …

Providing the first in-depth look at this breakthrough field, Combination Products includes practical guidelines and a detailed step-by-step process for the development of these novel technologies.

Smita Gopalaswamy, Venky Gopalaswamy, 2008

3

A Leadership Guide for Combination Fire Departments

This is the text that truly prepares fire inspectors for the real world.

Fred C. Windisch, Fred C. Crosby, 2007

4

Antiepileptic Drugs: Combination Therapy and Interactions

This book reviews the use of antiepileptic drugs focusing on the interactions between these drugs and between antiepileptics and other drugs. These interactions can be beneficial or can cause harm.

5

A Dangerous Combination: Living with a Bipolar Alcoholic

While only 2 percent of people are affected with bipolar disorder, 46 percent of them also deal with alcohol abuse, and my husband had that Dangerous Combination.

Sandy Derouin, Stephanie Dirocco, 2011

6

Hurdle Technologies: Combination Treatments for Food …

However, Hurdle Technologies: Combination Treatments for Food Stability, Safety and Quality is the first work on hurdle technology in which all aspects, the possibilities and limitations of hurdle technology, are comprehensively outlined …

Lothar Leistner, Grahame W. Gould, 2002

7

Development and Approval of Combination Products: A …

A step-by-step, integrated approach for successful, FDA-approved combination drug products Using a proven integrated approach to combination drug development, this book guides you step by step through all the preclinical, clinical, and …

8

Advances in Combination Therapy for Asthma and COPD

This volume provides specialists in respiratory medicine the most up to date information on the agents and discusses how they might be deployed in combination.

Jan Lotvall, William W. Busse, 2011

In this book we categorize the basic motifs, giving a plethora of fascintating and instructive examples for each type.

10

Combination Treatment in Autoimmune Diseases

This book constitutes the refereed proceedings of the 7th International Conference on Software Reuse, ICSR-7, held in Austin, Texas, USA, in April 2002.

W.B. Harrison, B.A.C. Dijkmans, 2002

10 NEWS ITEMS WHICH INCLUDE THE TERM «COMBINATION»

Find out what the national and international press are talking about and how the term combination is used in the context of the following news items.

Target and CVS Make for a Healthy Business Combination

NEW YORK (TheStreet) — Target (TGT — Get Report) and CVS Health (CVS — Get Report) signed a deal that would allow CVS to take over the … «TheStreet.com, Jul 15»

The Whiskey Daisy: A winning combination

The Whiskey Daisy is the official drink of the 21st running of the Grade II $500,000-added Indiana Derby on Saturday, July 18, 2015. Make your … «Indianapolis Star, Jul 15»

Combination of conditions reduce lifespan more than well-known …

(ABC News) — Smoking cigarettes, excess drinking, and diseases like HIV shorten a person’s lifespan, but other things may be taking away … «WTNH Connecticut News, Jul 15»

Impact Selector and Wireline Engineering announce strategic …

The combination brings together industry leaders in the design, development and deployment of the most technically-innovative products and … «WorldOil, Jul 15»

Heart disease and diabetes combination could shorten your life by a …

A combination of heart disease and diabetes can shorten your life by more than a decade, research has shown. Scientists came to the … «Western Daily Press, Jul 15»

Oregon Football Analysis: The Lethal Run-Pass Combination

… any play in the playbook on demand. The use of the Run-Pass Combination enables the Ducks to easily disguise the intent of a specific play. «FishDuck, Jul 15»

WP Carey Inc: An Attractive Combination Of Yield And Dividend …

W.P. Carey Inc (NYSE:WPC) is a global real estate investment trust that provides services including long-term sale-leaseback and build-to-suit … «Smarter Analyst, Jul 15»

FDA Approves Valsartan/Sacubitril (Entresto) Combination for Heart …

BETHESDA, MD — The US Food and Drug Administration (FDA) has now approved the combination tablet valsartan/sacubitril (Entresto, … «Medscape, Jul 15»

COMMENT: Denison, Fission combination to create diversified …

Denison Mines Corp. of Toronto and Fission Uranium Corp. of Kelowna, BC, have announced plans to combine their businesses, creating a … «Canadian Mining Journal, Jul 15»

Exercise and Healthy Aging: An Essential Combination

Exercise and Healthy Aging: An Essential Combination. By Jeffrey Tucker, DC, DACRB. Aging is inevitable, but how you age is not. Common physical signs of … «To Your Health, Jul 15»

REFERENCE

« EDUCALINGO. Combination [online]. Available <https://educalingo.com/en/dic-en/combination>. Apr 2023 ».

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Discover all that is hidden in the words on educalingo

What does Combination Mean?

Definitions

Definition as Noun
  • a collection of things that have been combined; an assemblage of separate parts or qualities
  • a coordinated sequence of chess moves
  • a sequence of numbers or letters that opens a combination lock
  • a group of people (often temporary) having a common purpose
  • an alliance of people or corporations or countries for a special purpose (formerly to achieve some antisocial end but now for general political or economic purposes)
  • the act of arranging elements into specified groups without regard to order
  • the act of combining things to form a new whole

Examples

  • «he forgot the combination to the safe»
  • «they were a winning combination«

Part of Speech

See also

often attributive

1

a

: a result or product of combining

Water is a combination of hydrogen and oxygen.

especially

: an alliance of individuals, corporations, or states united to achieve a social, political, or economic end

b

: two or more persons working as a team

Together they are a winning combination.

2

: an ordered sequence: such as

a

: a sequence of letters or numbers chosen in setting a lock

What is the combination to the safe?

also

: the mechanism operating or moved by the sequence

b

: a rapid sequence of punches in boxing

c

: any subset of a set considered without regard to order within the subset

3

: any of various one-piece undergarments for the upper and lower parts of the body

usually used in plural

wore combinations in the winter to keep warm

4

: an instrument designed to perform two or more tasks

The tool is a jackknife and bottle opener combination.

5

a

: the act or process of combining

especially

: that of uniting to form a chemical compound

The combination of these two chemicals can cause an explosion.

b

: the quality or state of being combined

Synonyms

Example Sentences



Water is a combination of hydrogen and oxygen.



The car’s interior is available in various color combinations.



A combination of factors led to her decision.



The milkshakes come in chocolate, strawberry, and vanilla, or any combination of these.



He has the right combination of talent and experience.



The combination of these two chemicals can cause an explosion.



What is the combination to the safe?

See More

Recent Examples on the Web

View Photos Inside, the Type S offers front seats with a bit of extra bolstering that are trimmed in a leather and suede upholstery combination available in either red, black, or white.


Joey Capparella, Car and Driver, 12 Apr. 2023





The box is made of plastic and polypropylene resin, a low-maintenance combination that can withstand the elements.


Jamie Weissman, Better Homes & Gardens, 11 Apr. 2023





Individual letters, or letter combinations, are processed separately and then integrated to form a coherent word.


Jamie Dickman, Popular Science, 11 Apr. 2023





Before Prince Louis, Prince George was known for his shorts and knee socks combinations — even in the winter.


Stephanie Petit, Peoplemag, 11 Apr. 2023





These 8-inch by 10-inch photos are framed with a light wood much like driftwood at the beach, and the soothing combination of gray and beige in every image will deliver a subtle calming effect within your living room or bedroom.


Merrell Readman, Travel + Leisure, 11 Apr. 2023





The combination of factors has made this the most expensive first-round series for Warriors’ tickets since 2013, when the team started its long run.


Sam Whiting, San Francisco Chronicle, 11 Apr. 2023





The two-pill combination has been considered safe and effective for inducing abortions for more than 20 years.


Amanda Gokee, BostonGlobe.com, 11 Apr. 2023





Advertisement Media folks welcome delight interesting stories, and for intrigue, none of MLB’s other 29 clubs can match the ‘23 Pads’ combination of star power and snooze-proof narrative.


Tom Krasovic, San Diego Union-Tribune, 10 Apr. 2023



See More

These examples are programmatically compiled from various online sources to illustrate current usage of the word ‘combination.’ Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

First Known Use

14th century, in the meaning defined at sense 1a

Time Traveler

The first known use of combination was
in the 14th century

Dictionary Entries Near combination

Cite this Entry

“Combination.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/combination. Accessed 14 Apr. 2023.

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More from Merriam-Webster on combination

Last Updated:
14 Apr 2023
— Updated example sentences

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Merriam-Webster unabridged

This article is about the mathematics of selecting part of a collection. For other uses, see Combination (disambiguation).

In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a kcombination of a set S is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has n elements, the number of k-combinations, denoted by C(n,k) or C_{k}^{n}, is equal to the binomial coefficient

{displaystyle {binom {n}{k}}={frac {n(n-1)dotsb (n-k+1)}{k(k-1)dotsb 1}},}

which can be written using factorials as textstyle {frac {n!}{k!(n-k)!}} whenever kleq n, and which is zero when k>n. This formula can be derived from the fact that each k-combination of a set S of n members has k! permutations so {displaystyle P_{k}^{n}=C_{k}^{n}times k!} or {displaystyle C_{k}^{n}=P_{k}^{n}/k!}.[1] The set of all k-combinations of a set S is often denoted by {displaystyle textstyle {binom {S}{k}}}.

A combination is a combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-combination with repetition, k-multiset,[2] or k-selection,[3] are often used.[4] If, in the above example, it were possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.

Although the set of three fruits was small enough to write a complete list of combinations, this becomes impractical as the size of the set increases. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.

Number of k-combinations[edit]

3-element subsets of a 5-element set

The number of k-combinations from a given set S of n elements is often denoted in elementary combinatorics texts by C(n,k), or by a variation such as C_{k}^{n}, {}_{n}C_{k}, {}^{n}C_{k}, C_{{n,k}} or even C_{n}^{k} (the last form is standard in French, Romanian, Russian, Chinese[5][6] and Polish texts[citation needed]). The same number however occurs in many other mathematical contexts, where it is denoted by {tbinom {n}{k}} (often read as «n choose k«); notably it occurs as a coefficient in the binomial formula, hence its name binomial coefficient. One can define {tbinom {n}{k}} for all natural numbers k at once by the relation

{displaystyle (1+X)^{n}=sum _{kgeq 0}{binom {n}{k}}X^{k},}

from which it is clear that

{displaystyle {binom {n}{0}}={binom {n}{n}}=1,}

and further,

{displaystyle {binom {n}{k}}=0}

for k > n.

To see that these coefficients count k-combinations from S, one can first consider a collection of n distinct variables Xs labeled by the elements s of S, and expand the product over all elements of S:

{displaystyle prod _{sin S}(1+X_{s});}

it has 2n distinct terms corresponding to all the subsets of S, each subset giving the product of the corresponding variables Xs. Now setting all of the Xs equal to the unlabeled variable X, so that the product becomes (1 + X)n, the term for each k-combination from S becomes Xk, so that the coefficient of that power in the result equals the number of such k-combinations.

Binomial coefficients can be computed explicitly in various ways. To get all of them for the expansions up to (1 + X)n, one can use (in addition to the basic cases already given) the recursion relation

{displaystyle {binom {n}{k}}={binom {n-1}{k-1}}+{binom {n-1}{k}},}

for 0 < k < n, which follows from (1 + X)n = (1 + X)n − 1(1 + X); this leads to the construction of Pascal’s triangle.

For determining an individual binomial coefficient, it is more practical to use the formula

{displaystyle {binom {n}{k}}={frac {n(n-1)(n-2)cdots (n-k+1)}{k!}}.}

The numerator gives the number of k-permutations of n, i.e., of sequences of k distinct elements of S, while the denominator gives the number of such k-permutations that give the same k-combination when the order is ignored.

When k exceeds n/2, the above formula contains factors common to the numerator and the denominator, and canceling them out gives the relation

{displaystyle {binom {n}{k}}={binom {n}{n-k}},}

for 0 ≤ kn. This expresses a symmetry that is evident from the binomial formula, and can also be understood in terms of k-combinations by taking the complement of such a combination, which is an (nk)-combination.

Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember:

{displaystyle {binom {n}{k}}={frac {n!}{k!(n-k)!}},}

where n! denotes the factorial of n. It is obtained from the previous formula by multiplying denominator and numerator by (nk)!, so it is certainly computationally less efficient than that formula.

The last formula can be understood directly, by considering the n! permutations of all the elements of S. Each such permutation gives a k-combination by selecting its first k elements. There are many duplicate selections: any combined permutation of the first k elements among each other, and of the final (n − k) elements among each other produces the same combination; this explains the division in the formula.

From the above formulas follow relations between adjacent numbers in Pascal’s triangle in all three directions:

{displaystyle {binom {n}{k}}={begin{cases}{binom {n}{k-1}}{frac {n-k+1}{k}}&quad {text{if }}k>0\{binom {n-1}{k}}{frac {n}{n-k}}&quad {text{if }}k<n\{binom {n-1}{k-1}}{frac {n}{k}}&quad {text{if }}n,k>0end{cases}}.}

Together with the basic cases {tbinom {n}{0}}=1={tbinom {n}{n}}, these allow successive computation of respectively all numbers of combinations from the same set (a row in Pascal’s triangle), of k-combinations of sets of growing sizes, and of combinations with a complement of fixed size nk.

Example of counting combinations[edit]

As a specific example, one can compute the number of five-card hands possible from a standard fifty-two card deck as:[7]

{displaystyle {52 choose 5}={frac {52times 51times 50times 49times 48}{5times 4times 3times 2times 1}}={frac {311{,}875{,}200}{120}}=2{,}598{,}960.}

Alternatively one may use the formula in terms of factorials and cancel the factors in the numerator against parts of the factors in the denominator, after which only multiplication of the remaining factors is required:

{displaystyle {begin{alignedat}{2}{52 choose 5}&={frac {52!}{5!47!}}\[5pt]&={frac {52times 51times 50times 49times 48times {cancel {47!}}}{5times 4times 3times 2times {cancel {1}}times {cancel {47!}}}}\[5pt]&={frac {52times 51times 50times 49times 48}{5times 4times 3times 2}}\[5pt]&={frac {(26times {cancel {2}})times (17times {cancel {3}})times (10times {cancel {5}})times 49times (12times {cancel {4}})}{{cancel {5}}times {cancel {4}}times {cancel {3}}times {cancel {2}}}}\[5pt]&={26times 17times 10times 49times 12}\[5pt]&=2{,}598{,}960.end{alignedat}}}

Another alternative computation, equivalent to the first, is based on writing

{displaystyle {n choose k}={frac {(n-0)}{1}}times {frac {(n-1)}{2}}times {frac {(n-2)}{3}}times cdots times {frac {(n-(k-1))}{k}},}

which gives

{displaystyle {52 choose 5}={frac {52}{1}}times {frac {51}{2}}times {frac {50}{3}}times {frac {49}{4}}times {frac {48}{5}}=2{,}598{,}960.}

When evaluated in the following order, 52 ÷ 1 × 51 ÷ 2 × 50 ÷ 3 × 49 ÷ 4 × 48 ÷ 5, this can be computed using only integer arithmetic. The reason is that when each division occurs, the intermediate result that is produced is itself a binomial coefficient, so no remainders ever occur.

Using the symmetric formula in terms of factorials without performing simplifications gives a rather extensive calculation:

{displaystyle {begin{aligned}{52 choose 5}&={frac {n!}{k!(n-k)!}}={frac {52!}{5!(52-5)!}}={frac {52!}{5!47!}}\[6pt]&={tfrac {80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000}{120times 258,623,241,511,168,180,642,964,355,153,611,979,969,197,632,389,120,000,000,000}}\[6pt]&=2{,}598{,}960.end{aligned}}}

Enumerating k-combinations[edit]

One can enumerate all k-combinations of a given set S of n elements in some fixed order, which establishes a bijection from an interval of {tbinom {n}{k}} integers with the set of those k-combinations. Assuming S is itself ordered, for instance S = { 1, 2, …, n }, there are two natural possibilities for ordering its k-combinations: by comparing their smallest elements first (as in the illustrations above) or by comparing their largest elements first. The latter option has the advantage that adding a new largest element to S will not change the initial part of the enumeration, but just add the new k-combinations of the larger set after the previous ones. Repeating this process, the enumeration can be extended indefinitely with k-combinations of ever larger sets. If moreover the intervals of the integers are taken to start at 0, then the k-combination at a given place i in the enumeration can be computed easily from i, and the bijection so obtained is known as the combinatorial number system. It is also known as «rank»/»ranking» and «unranking» in computational mathematics.[8][9]

There are many ways to enumerate k combinations. One way is to visit all the binary numbers less than 2n. Choose those numbers having k nonzero bits, although this is very inefficient even for small n (e.g. n = 20 would require visiting about one million numbers while the maximum number of allowed k combinations is about 186 thousand for k = 10). The positions of these 1 bits in such a number is a specific k-combination of the set { 1, …, n }.[10] Another simple, faster way is to track k index numbers of the elements selected, starting with {0 .. k−1} (zero-based) or {1 .. k} (one-based) as the first allowed k-combination and then repeatedly moving to the next allowed k-combination by incrementing the last index number if it is lower than n-1 (zero-based) or n (one-based) or the last index number x that is less than the index number following it minus one if such an index exists and resetting the index numbers after x to {x+1, x+2, …}.

Number of combinations with repetition[edit]

A kcombination with repetitions, or kmulticombination, or multisubset of size k from a set S of size n is given by a set of k not necessarily distinct elements of S, where order is not taken into account: two sequences define the same multiset if one can be obtained from the other by permuting the terms. In other words, it is a sample of k elements from a set of n elements allowing for duplicates (i.e., with replacement) but disregarding different orderings (e.g. {2,1,2} = {1,2,2}). Associate an index to each element of S and think of the elements of S as types of objects, then we can let x_{i} denote the number of elements of type i in a multisubset. The number of multisubsets of size k is then the number of nonnegative integer (so allowing zero) solutions of the Diophantine equation:[11]

{displaystyle x_{1}+x_{2}+ldots +x_{n}=k.}

If S has n elements, the number of such k-multisubsets is denoted by

{displaystyle left(!!{binom {n}{k}}!!right),}

a notation that is analogous to the binomial coefficient which counts k-subsets. This expression, n multichoose k,[12] can also be given in terms of binomial coefficients:

{displaystyle left(!!{binom {n}{k}}!!right)={binom {n+k-1}{k}}.}

This relationship can be easily proved using a representation known as stars and bars.[13]

Bijection between 3-subsets of a 7-set (left) and 3-multisets with elements from a 5-set (right).
This illustrates that {textstyle {binom {7}{3}}=left(!!{binom {5}{3}}!!right)}.

As with binomial coefficients, there are several relationships between these multichoose expressions. For example, for ngeq 1,kgeq 0,

{displaystyle left(!!{binom {n}{k}}!!right)=left(!!{binom {k+1}{n-1}}!!right).}

This identity follows from interchanging the stars and bars in the above representation.[15]

Example of counting multisubsets[edit]

For example, if you have four types of donuts (n = 4) on a menu to choose from and you want three donuts (k = 3), the number of ways to choose the donuts with repetition can be calculated as

{displaystyle left(!!{binom {4}{3}}!!right)={binom {4+3-1}{3}}={binom {6}{3}}={frac {6times 5times 4}{3times 2times 1}}=20.}

This result can be verified by listing all the 3-multisubsets of the set S = {1,2,3,4}. This is displayed in the following table.[16] The second column lists the donuts you actually chose, the third column shows the nonnegative integer solutions [x_{1},x_{2},x_{3},x_{4}] of the equation x_{1}+x_{2}+x_{3}+x_{4}=3 and the last column gives the stars and bars representation of the solutions.[17]

No. 3-multiset Eq. solution Stars and bars
1 {1,1,1} [3,0,0,0] bigstar bigstar bigstar |||
2 {1,1,2} [2,1,0,0] bigstar bigstar |bigstar ||
3 {1,1,3} [2,0,1,0] bigstar bigstar ||bigstar |
4 {1,1,4} [2,0,0,1] bigstar bigstar |||bigstar
5 {1,2,2} [1,2,0,0] bigstar |bigstar bigstar ||
6 {1,2,3} [1,1,1,0] bigstar |bigstar |bigstar |
7 {1,2,4} [1,1,0,1] bigstar |bigstar ||bigstar
8 {1,3,3} [1,0,2,0] bigstar ||bigstar bigstar |
9 {1,3,4} [1,0,1,1] bigstar ||bigstar |bigstar
10 {1,4,4} [1,0,0,2] bigstar |||bigstar bigstar
11 {2,2,2} [0,3,0,0] |bigstar bigstar bigstar ||
12 {2,2,3} [0,2,1,0] |bigstar bigstar |bigstar |
13 {2,2,4} [0,2,0,1] |bigstar bigstar ||bigstar
14 {2,3,3} [0,1,2,0] |bigstar |bigstar bigstar |
15 {2,3,4} [0,1,1,1] |bigstar |bigstar |bigstar
16 {2,4,4} [0,1,0,2] |bigstar ||bigstar bigstar
17 {3,3,3} [0,0,3,0] ||bigstar bigstar bigstar |
18 {3,3,4} [0,0,2,1] ||bigstar bigstar |bigstar
19 {3,4,4} [0,0,1,2] ||bigstar |bigstar bigstar
20 {4,4,4} [0,0,0,3] |||bigstar bigstar bigstar

Number of k-combinations for all k[edit]

The number of k-combinations for all k is the number of subsets of a set of n elements. There are several ways to see that this number is 2n. In terms of combinations, {textstyle sum _{0leq {k}leq {n}}{binom {n}{k}}=2^{n}}, which is the sum of the nth row (counting from 0) of the binomial coefficients in Pascal’s triangle. These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2n − 1, where each digit position is an item from the set of n.

Given 3 cards numbered 1 to 3, there are 8 distinct combinations (subsets), including the empty set:

{displaystyle |{{};{1};{2};{1,2};{3};{1,3};{2,3};{1,2,3}}|=2^{3}=8}

Representing these subsets (in the same order) as base 2 numerals:

  • 0 – 000
  • 1 – 001
  • 2 – 010
  • 3 – 011
  • 4 – 100
  • 5 – 101
  • 6 – 110
  • 7 – 111

Probability: sampling a random combination[edit]

There are various algorithms to pick out a random combination from a given set or list. Rejection sampling is extremely slow for large sample sizes. One way to select a k-combination efficiently from a population of size n is to iterate across each element of the population, and at each step pick that element with a dynamically changing probability of {textstyle {frac {k-#{text{samples chosen}}}{n-#{text{samples visited}}}}} (see Reservoir sampling). Another is to pick a random non-negative integer less than textstyle {binom  nk} and convert it into a combination using the combinatorial number system.

Number of ways to put objects into bins[edit]

A combination can also be thought of as a selection of two sets of items: those that go into the chosen bin and those that go into the unchosen bin. This can be generalized to any number of bins with the constraint that every item must go to exactly one bin. The number of ways to put objects into bins is given by the multinomial coefficient

{displaystyle {n choose k_{1},k_{2},ldots ,k_{m}}={frac {n!}{k_{1}!,k_{2}!cdots k_{m}!}},}

where n is the number of items, m is the number of bins, and k_{i} is the number of items that go into bin i.

One way to see why this equation holds is to first number the objects arbitrarily from 1 to n and put the objects with numbers {displaystyle 1,2,ldots ,k_{1}} into the first bin in order, the objects with numbers {displaystyle k_{1}+1,k_{1}+2,ldots ,k_{2}} into the second bin in order, and so on. There are n! distinct numberings, but many of them are equivalent, because only the set of items in a bin matters, not their order in it. Every combined permutation of each bins’ contents produces an equivalent way of putting items into bins. As a result, every equivalence class consists of {displaystyle k_{1}!,k_{2}!cdots k_{m}!} distinct numberings, and the number of equivalence classes is {displaystyle textstyle {frac {n!}{k_{1}!,k_{2}!cdots k_{m}!}}}.

The binomial coefficient is the special case where k items go into the chosen bin and the remaining n-k items go into the unchosen bin:

{displaystyle {binom {n}{k}}={n choose k,n-k}={frac {n!}{k!(n-k)!}}.}

See also[edit]

  • Binomial coefficient
  • Combinatorics
  • Block design
  • Kneser graph
  • List of permutation topics
  • Multiset
  • Pascal’s triangle
  • Permutation
  • Probability
  • Subset

Notes[edit]

  1. ^ Reichl, Linda E. (2016). «2.2. Counting Microscopic States». A Modern Course in Statistical Physics. WILEY-VCH. p. 30. ISBN 978-3-527-69048-0.
  2. ^ Mazur 2010, p. 10
  3. ^ Ryser 1963, p. 7 also referred to as an unordered selection.
  4. ^ When the term combination is used to refer to either situation (as in (Brualdi 2010)) care must be taken to clarify whether sets or multisets are being discussed.
  5. ^ High School Textbook for full-time student (Required) Mathematics Book II B (in Chinese) (2nd ed.). China: People’s Education Press. June 2006. pp. 107–116. ISBN 978-7-107-19616-4.
  6. ^ 人教版高中数学选修2-3 (Mathematics textbook, volume 2-3, for senior high school, People’s Education Press). People’s Education Press. p. 21.
  7. ^ Mazur 2010, p. 21
  8. ^ Lucia Moura. «Generating Elementary Combinatorial Objects» (PDF). Site.uottawa.ca. Archived (PDF) from the original on 9 October 2022. Retrieved 10 April 2017.
  9. ^ «SAGE : Subsets» (PDF). Sagemath.org. Retrieved 10 April 2017.
  10. ^ «Combinations — Rosetta Code». 23 October 2022.[user-generated source?]
  11. ^ Brualdi 2010, p. 52
  12. ^ Benjamin & Quinn 2003, p. 70
  13. ^ In the article Stars and bars (combinatorics) the roles of n and k are reversed.
  14. ^ Benjamin & Quinn 2003, pp. 71 –72
  15. ^ Benjamin & Quinn 2003, p. 72 (identity 145)
  16. ^ Benjamin & Quinn 2003, p. 71
  17. ^ Mazur 2010, p. 10 where the stars and bars are written as binary numbers, with stars = 0 and bars = 1.

References[edit]

  • Benjamin, Arthur T.; Quinn, Jennifer J. (2003), Proofs that Really Count: The Art of Combinatorial Proof, The Dolciani Mathematical Expositions 27, The Mathematical Association of America, ISBN 978-0-88385-333-7
  • Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Pearson Prentice Hall, ISBN 978-0-13-602040-0
  • Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, INC, 1999.
  • Mazur, David R. (2010), Combinatorics: A Guided Tour, Mathematical Association of America, ISBN 978-0-88385-762-5
  • Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs 14, Mathematical Association of America

External links[edit]

  • Topcoder tutorial on combinatorics
  • C code to generate all combinations of n elements chosen as k
  • Many Common types of permutation and combination math problems, with detailed solutions
  • The Unknown Formula For combinations when choices can be repeated and order does not matter
  • Combinations with repetitions (by: Akshatha AG and Smitha B)[permanent dead link]
  • The dice roll with a given sum problem An application of the combinations with repetition to rolling multiple dice

noun

COLLOCATIONS FROM OTHER ENTRIES

a colour combination/combination of colours (=the colours that exist or that you put together)

▪ In autumn the leaves create lovely colour combinations.

a colour combination/combination of colours (=the colours that exist or that you put together)

▪ In autumn the leaves create lovely colour combinations.

a combination/variety/number of factors

▪ A combination of factors led to the closure of the factory.

a lethal combination

▪ Higher taxes and higher inflation were a lethal combination.

a set/combination of circumstances

▪ This was a very unusual set of circumstances.

combination lock

combination oven

COLLOCATIONS FROM CORPUS

■ ADJECTIVE

different

▪ The system can be used in different combinations and packs down into one stuff sack.

▪ Abbott has not yet analyzed the results for patients taking different combinations of drugs.

▪ Taken together, different combinations of presidential, electoral, and party systems can have profound effects on democratic performance.

▪ Why are there different combinations of extinct species in each layer of rock?

▪ You end up with a gridded image with each square a different combination of negative exposure and flash exposure.

▪ He mixes different combinations and succeeds in getting the yellow color and then losing it.

▪ You will in fact find any shade will look acceptable so experiment with different combinations before embarking on an actual garment.

▪ It is a random arrangement of all the colors imaginable, with continually different color combinations at every glance.

lethal

▪ N New tank syndrome: A lethal combination of all the problems mentioned in this article.

▪ Not to mention that lethal combination of hands and strength.

▪ Together however, these factors can prove a lethal combination.

▪ This is a potentially lethal combination, but modern techniques help to safeguard us from those risks.

▪ The little church has suffered from the rigours of time and town planning, normally a lethal combination, and survived.

▪ An aptly named gas, discovered in 1766, it had that potentially lethal combination of having both explosive and lifting properties.

possible

▪ This distinction, coupled with the previous one, yields four possible combinations of fact, each needing discussion.

▪ The design of an experiment typically takes two of these factors and combines them in all possible combinations.

▪ Some people fondly believe that chess-playing computers work by internally trying out all possible combinations of chess moves.

▪ There are 25 possible combinations of two or more of the liquids.

▪ Now let us look at the possible combinations.

▪ As you can imagine, there are many possible combinations.

▪ The figure therefore gives the possible combinations of project outcome and additional information.

▪ The Census forms admit of some 63 possible combinations of race.

right

▪ Always practise on a sample of fabric first to find the right combination.

▪ The right combination of convictions could ship him to death row.

▪ It is sometimes difficult to fet the right combination of storage and hanging space.

▪ Picking the right combination for a team is one of the keys to a successful climb.

▪ Systematic Cleaning Systematic cleaning is the adoption of the right combination of energy for the circumstances with regard to time and cost.

▪ The makers Schott claim that this porous glass material has exactly the right combination of aerobic and anaerobic opportunities for bacterial growth.

unique

▪ From this superb range comes Ribbon Gel; a unique striped combination gel.

▪ They employed a unique combination of theoretic analysis, practical grasp, and imaginative foresight.

▪ Our unique combination of skills has helped literally thousands of companies.

▪ The scenery is breathtaking, a unique combination of mountains, fells, pastures and forested valleys filled with lakes and farms.

▪ A unique combination, offering an unequalled nationwide service.

▪ The membership of the teams is bringing together a unique combination of experience and expertise in developing services for carers.

▪ The Open University has a unique combination of foundation courses followed by a modular course structure.

▪ This may well be a unique combination which will produce a slightly different organism with new characteristics.

various

▪ The point is that various combinations of government and private sector organization, finance and production are possible.

▪ The manager is concerned about how teams are orchestrated on various scales, with various combinations of players and various resources exchanged.

▪ Other techniques may also be used, both in isolation and in various combinations.

▪ These are doubled up in various combinations to form up to 64 hexagrams.

▪ Yamaha use various combinations of polyester and polyurethane on their guitars.

▪ This approach is broadly statistical in nature, as it involves corpus analysis to determine the empirical likelihood of various syntactic combinations.

▪ The results of associating various combinations of thymocytes and stromal cells are summarized in Table 1.

▪ The remaining lines to the right have all been climbed by various combinations of Newton, McGinley and Liz Wade.

■ NOUN

drug

▪ Nineteen had used only one drug prior to heroin use, principally cannabis, the rest using a variety of drug combinations.

▪ At that point, they are put on other drug combinations.

lock

▪ He laid it on the bed, turned the combination locks and lifted the lid.

▪ I took it down from the wall to reveal the circular door of a wall safe with a combination lock.

▪ Next to it was a steel security cupboard fitted with a combination lock.

▪ The case sported matching brass hardware, including a pair of clasps, each with its own three-number combination lock.

▪ Remember the analogy of the combination lock.

▪ The only thing holding me up would be fumbling at the combination lock or renting a towel.

▪ And a briefcase with one of those combination locks.

▪ I had twenty minutes to see what I could do about his combination locks.

therapy

▪ But he says that any drugs to block this process would have to form part of a combination therapy.

▪ Kaiser says it will also provide the latest three-drug combination therapies, if appropriate.

▪ Data from combination therapy studies are also gradually becoming available.

▪ Thus, for several reasons, combination therapy with artemisinin or a derivative makes therapeutic sense.

▪ She said he had not shown any symptoms for some time but had recently started taking combination therapy drugs.

■ VERB

involve

▪ Training methods involve repeating these combinations until the juxtaposition of foot, hip and shoulder becomes automatic.

▪ The program involves a combination of relaxation and visualization exercises, along with some structured statements called affirmations.

▪ Her exercising and her diets, involving theories about food combinations, had irritated me for months.

▪ Frequently the stresses involve a combination of these different levels.

▪ Helping the Disabled Siting a new bureau involves a complex combination of local objectives, suitability, costing and availability.

▪ Human machines everywhere are built up from mechanisms involving combinations of motion in a straight line and motion in a circle.

▪ The approach to pro-active searches is well established and involves a combination of desk and field research.

offer

▪ The latest addition to our successful twin-colour Designer Range of Tilturn and Casement windows offers endless colour combinations, inside and out.

▪ Technology analysts offered a combination of answers last weekend.

▪ Powter offers women an irresistible combination of promises.

▪ The approach is cognitive and the books offer a combination of explanations, examples, and a variety of imaginative exercises.

▪ Prestige saucepans and bakeware offer the perfect combination — excellent performance and quality.

▪ Delapre Park offers a charming combination of an Elizabethan walled garden, parkland and woodland walks.

▪ A well-designed and efficiently controlled system offers the best combination of comfort, convenience and economy, but it’s not essential.

use

▪ Many complementary practitioners use combinations of these disciplines.

▪ We have arbitrarily grouped them in three categories: Many of these methods can be used in combination.

▪ Thrusters were then used to pull the combination further ahead.

▪ We shall only use these strange complex-number combinations at the quantum level.

▪ Different tribes often use different colours and combinations of feathers to identify themselves.

▪ Sometimes it is necessary to use a combination of two AEDs.

▪ She used a combination of small weights, to have the fun of adding them up.

▪ They are used in combination to set apart two main groups, the detrital and the chemical.

EXAMPLES FROM OTHER ENTRIES

▪ Banana, orange juice, and cream may seem an odd combination, but together they make a delicious drink.

▪ I’ll have the shrimp and chicken combination, please.

▪ If all the team are playing well, then don’t change a winning combination.

▪ Our problems were due to a combination of bad management and lack of experience.

▪ The new device is a combination copier, fax, image scanner, and document printer.

▪ The sweater is made of a combination of natural and synthetic fibres.

▪ Their music is an odd combination of jazz and opera.

▪ They were a perfect combination — Anton as chef and Guy as restaurant manager.

▪ What’s the matter? Can’t you remember the combination?

EXAMPLES FROM CORPUS

▪ A combination of fears leads to this unfortunate situation.

▪ He had tried every combination of the antibiotics we had on board, and was running out of tablets.

▪ His leadership style was to maintain power through a combination of force and cunning strategy.

▪ Most were attracted by her intoxicating combination of beauty, wit and charm, but others were cowed.

▪ The nationalization of local politics arose from a specific combination of economic, social and political processes which no longer applies.

▪ This combination of your political beliefs and actions is the essence of the domain of political science called political behavior or micropolitics.

▪ This, as far as we know, is the first reported case of the combination of gold induced and infective colitis.

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