Combination

From Wikipedia, the free encyclopedia

For other uses, see Combination (disambiguation).

In combinatorial mathematics, a combination is an un-ordered collection of unique elements. (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 because combinations are defined by the elements contained in them, so the set {1, 1, 1} is the same as {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

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

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

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

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

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

Example:

${70 \choose 4} = \frac { 70 }{ 4 } \times \frac { 69 }{ 3 } \times \frac { 68 }{ 2 } \times \frac { 67 }{ 1 } = 916895.$

[edit] How to use with a calculator

Most calculators have an nCr key. In most advanced desktop calculators, however, the key is hidden. Example: in TI-83, press MATH, right three times, and press 3.

Simply enter n,nCr,k =.