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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

permutation
s). 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 ''k''-combination 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 as C^n_k, is equal to the
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
: \binom nk = \frac, which can be written using
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s as \textstyle\frac whenever k\leq 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

permutations
so P^n_k = C^n_k \times k! or C^n_k = P^n_k / k!. The set of all ''k''-combinations of a set ''S'' is often denoted by \textstyle\binom Sk. 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''-selection, ''k''-
multiset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, or ''k''-combination with repetition are often used. 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 In poker Poker is a family of card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules in ways similar List of poker hands, to these rankings. While the earli ...
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

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^n_k, _nC_k, ^nC_k, C_ or even C_n^k (the latter form is standard in French, Romanian, Russian, Chinese and Polish texts). The same number however occurs in many other mathematical contexts, where it is denoted by \tbinom nk (often read as "''n'' choose ''k''"); notably it occurs as a coefficient in the
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of exponentiation, powers of a binomial (polynomial), binomial. According to the theorem, it is possible to expand the polynomial into a summati ...
, hence its name binomial coefficient. One can define \tbinom nk for all natural numbers ''k'' at once by the relation :(1 + X)^n = \sum_\binom X^k, from which it is clear that :\binom = \binom = 1, and further, :\binom = 0 for ''k'' > ''n''. To see that these coefficients count ''k''-combinations from ''S'', one can first consider a collection of ''n'' distinct variables ''X''''s'' labeled by the elements ''s'' of ''S'', and expand the
product
product
over all elements of ''S'': :\prod_(1+X_s); it has 2''n'' distinct terms corresponding to all the subsets of ''S'', each subset giving the product of the corresponding variables ''X''''s''. Now setting all of the ''X''''s'' equal to the unlabeled variable ''X'', so that the product becomes , the term for each ''k''-combination from ''S'' becomes ''X''''k'', 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 , one can use (in addition to the basic cases already given) the recursion relation :\binom = \binom + \binom, for 0 < ''k'' < ''n'', which follows from =; this leads to the construction of
Pascal's triangle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. For determining an individual binomial coefficient, it is more practical to use the formula :\binom nk = \frac. The
numerator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
gives the number of ''k''-permutations of ''n'', i.e., of sequences of ''k'' distinct elements of ''S'', while the
denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
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 : \binom nk = \binom n, for 0 ≤ ''k'' ≤ ''n''. 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 A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
of such a combination, which is an -combination. Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember: : \binom nk = \frac, where ''n''! denotes the
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of ''n''. It is obtained from the previous formula by multiplying denominator and numerator by !, 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: : \binom nk = \begin \binom n \frac k &\quad \text k > 0 \\ \binom k \frac n &\quad \text k < n \\ \binom \frac nk &\quad \text n, k > 0 \end . Together with the basic cases \tbinom n0=1=\tbinom nn, 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 .


Example of counting combinations

As a specific example, one can compute the number of five-card hands possible from a standard fifty-two card deck as: : = \frac = \frac = 2598960. 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: :\begin &= \frac \\
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&= \frac \\
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&= \frac \\
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&= \frac \\
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&= \\
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&= 2598960. \end Another alternative computation, equivalent to the first, is based on writing : = \frac 1 \times \frac 2 \times \frac 3 \times \cdots \times \frac k, which gives : = \frac1 \times \frac2 \times \frac3 \times \frac4 \times \frac5 = 2598960. When evaluated in the following order, , 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: : \begin &= \frac = \frac = \frac \\
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&= \tfrac \\
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&= 2598960. \end


Enumerating ''k''-combinations

One can
enumerate Enumerate could refer to: * Enumeration, a mathematical, theoretical concept of an exhaustive listing of compatible items * Enumerate (project), a collaborative research project about digitization of cultural heritage * Enumerated type in computer p ...
all ''k''-combinations of a given set ''S'' of ''n'' elements in some fixed order, which establishes a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
from an interval of \tbinom nk integers with the set of those ''k''-combinations. Assuming ''S'' is itself ordered, for instance ''S'' = , 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 systemIn mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, is a correspondence between natural numbers (taken to include 0) ''N'' and '' ...
. It is also known as "rank"/"ranking" and "unranking" in computational mathematics. There are many ways to enumerate ''k'' combinations. One way is to visit all the binary numbers less than 2''n''. 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 . Another simple, faster way is to track ''k'' index numbers of the elements selected, starting with (zero-based) or (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 .


Number of combinations with repetition

A ''k''-combination with repetitions, or ''k''-multicombination, 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. = ). 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: :x_1 + x_2 + \ldots + x_n = k. If ''S'' has ''n'' elements, the number of such ''k''-multisubsets is denoted by, :\left(\!\!\binom\!\!\right), a notation that is analogous to the
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
which counts ''k''-subsets. This expression, ''n'' multichoose ''k'', can also be given in terms of binomial coefficients: :\left(\!\!\binom\!\!\right)=\binom. This relationship can be easily proved using a representation known as stars and bars. A solution of the above Diophantine equation can be represented by x_1 ''stars'', a separator (a ''bar''), then x_2 more stars, another separator, and so on. The total number of stars in this representation is ''k'' and the number of bars is ''n'' - 1 (since a separation into n parts needs n-1 separators). Thus, a string of ''k'' + ''n'' - 1 (or ''n'' + ''k'' - 1) symbols (stars and bars) corresponds to a solution if there are ''k'' stars in the string. Any solution can be represented by choosing ''k'' out of positions to place stars and filling the remaining positions with bars. For example, the solution x_1 = 3, x_2 = 2, x_3 = 0, x_4 = 5 of the equation x_1 + x_2 + x_3 + x_4 = 10 (''n'' = 4 and ''k'' = 10) can be represented by \bigstar \bigstar \bigstar , \bigstar \bigstar , , \bigstar \bigstar \bigstar \bigstar \bigstar. The number of such strings is the number of ways to place 10 stars in 13 positions, \binom = \binom = 286, which is the number of 10-multisubsets of a set with 4 elements. As with binomial coefficients, there are several relationships between these multichoose expressions. For example, for n \ge 1, k \ge 0, :\left(\!\!\binom\!\!\right)=\left(\!\!\binom\!\!\right). This identity follows from interchanging the stars and bars in the above representation.


Example of counting multisubsets

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 :\left(\!\!\binom\!\!\right) = \binom3 = \binom = \frac = 20. This result can be verified by listing all the 3-multisubsets of the set ''S'' = . This is displayed in the following table. The second column lists the donuts you actually chose, the third column shows the nonnegative integer solutions _1,x_2,x_3,x_4/math> 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. where the stars and bars are written as binary numbers, with stars = 0 and bars = 1.


Number of ''k''-combinations for all ''k''

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 2''n''. In terms of combinations, \sum_\binom n k = 2^n, which is the sum of the ''n''th row (counting from 0) of the
binomial coefficients , in which each entry is the sum of the two immediately above. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of inte ...
in
Pascal's triangle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. These combinations (subsets) are enumerated by the 1 digits of the set of
base 2 Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium's three mobile telephone company, telecommunications operators. It is a subsidiary of Telenet (Belgium), ...

base 2
numbers counting from 0 to 2''n'' − 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 In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are u ...

subsets
), including the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
: :, \, = 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

There are various
algorithms In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
to pick out a random combination from a given set or list.
Rejection samplingIn numerical analysis and computational statistics, rejection sampling is a basic technique used to generate observations from a distribution. It is also commonly called the acceptance-rejection method or "accept-reject algorithm" and is a type of ...
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 \frac (see
Reservoir sampling Reservoir sampling is a family of randomized algorithm A ''randomized algorithm'' is an algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B ...
). Another is to pick a random non-negative integer less than \textstyle\binom nk and convert it into a combination using the
combinatorial number systemIn mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, is a correspondence between natural numbers (taken to include 0) ''N'' and '' ...
.


Number of ways to put objects into bins

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 In mathematics, the multinomial theorem describes how to expand a power (mathematics), power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any posi ...
: = \frac, 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 1, 2, \ldots, k_1 into the first bin in order, the objects with numbers 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 k_1!\, k_2! \cdots k_m! distinct numberings, and the number of equivalence classes is \textstyle\frac. 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: : \binom nk = = \frac.


See also

*
Binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
*
Combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
*
Block design In combinatorial mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, ...
*
Kneser graph In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theo ...
* List of permutation topics *
Multiset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Pascal's triangle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
*
Permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

Permutation
*
Probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

Probability
*
Subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

Subset


Notes


References

* * *
Erwin Kreyszig Erwin O. Kreyszig (January 6, 1922 in Pirna, Germany – December 12, 2008) was a German Canadian applied mathematician and the Professor of Mathematics at Carleton University in Ottawa, Ottawa, Ontario, Canada. He was a pioneer in the field of ap ...
, ''Advanced Engineering Mathematics'', John Wiley & Sons, INC, 1999. * *


External links


Topcoder tutorial on combinatorics

C code to generate all combinations of n elements chosen as k


* ttp://www.murderousmaths.co.uk/books/unknownform.htm The Unknown FormulaFor combinations when choices can be repeated and order does ''not'' matter
Combinations with repetitions (by: Akshatha AG and Smitha B)
{Dead link, date=November 2019 , bot=InternetArchiveBot , fix-attempted=yes
The dice roll with a given sum problem
An application of the combinations with repetition to rolling multiple dice Combinatorics