Combination

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For other uses, see Combination (disambiguation).

In combinatorial mathematics, a combination is an un-ordered collection of unique sizes. (An ordered collection is called a permutation.) Given S, the set of all possible unique elements, a combination is a subset of the elements of S. The order of the elements in a combination is not important (two lists with the same elements in different orders are considered to be the same combination). Also, the elements cannot be repeated in a combination (every element appears uniquely once); this is often referred to as "without replacement/repetition". This is because combinations are defined by the elements contained in them, thus the set {1,1,2} is the same as {2,1,1}. For example, from a 52-card deck any 5 cards can form a valid combination (a hand). The order of the cards doesn't matter and there can be no repetition of cards.

A k-combination (or k-subset) is a subset with k elements. The number of k-combinations (each of size k) from a set S with n elements (size n) is the binomial coefficient (also known as the "choose function"):

$C^n_k = {n \choose k} = \frac{n!}{k!(n-k)!}.$

where n is the number of objects from which you can choose and k is the number to be chosen, and $n!$ denotes the factorial.

As an example, the number of five-card hands possible from a standard fifty-two card deck is:

${52 \choose 5} = \frac{n!}{k!(n-k)!} = \frac{52!}{5!(52-5)!} = \frac{52!}{5!47!} = 2598960.$

The number of combinations with repetition can be calculated as:

${{(n + k - 1)!} \over {k!(n - 1)!}} = {{n + k - 1} \choose {k}} = {{n + k - 1} \choose {n - 1}}$

For example, if you have ten types of donuts (n) on a menu to choose from and you want three donuts (k) there are (10 + 3 − 1)! / 3!(10 − 1)! = 220 ways to choose (see also multiset).

A combination is a special case of a partition of a set; specifically, a partition into two sets of size k and n − k.

Since it is impractical to calculate $n!$ if the value of n is very large, a more efficient algorithm is

${n \choose k} = \frac { ( n - 0 ) }{ (k - 0) } \times \frac { ( n - 1 ) }{ (k - 1) } \times \frac { ( n - 2 ) }{ (k - 2) } \times \frac { ( n - 3 ) }{ (k - 3) } \times \cdots \times \frac { ( n - (k - 1) ) }{ (k - (k - 1)) }.$