Combination

Combinations are studied in combinatorics: let S be a set; the combinations of this set are its subsets. A k-combination is a subset of S with k elements. The order of listing the elements is not important in combinations: two lists with the same elements in different orders are considered to be the same combination. The number of k-combinations of set with n elements is the binomial coefficient "n choose k", written as _nC_k, ⁿC_k or as

<math>{n \choose k},</math>

or occasionally as C(n, k).

One method of deriving a formula for _nC_k proceeds as follows:

1. Count the number of ways in which one can make an ordered list of k different elements from the set of n. This is equivalent to calculating the number of k-permutations.
2. Recognizing that we have listed every subset many times, we correct the calculation by dividing by the number of different lists containing the same k elements:

<math> {n \choose k} = \frac{P(n,k)}{P(k,k)} </math>

Since

<math> P(n,k) = \frac{n!}{(n-k)!} </math>

(see factorial), we find

<math> {n \choose k} = \frac{n!}{k! \cdot (n-k)!} </math>

It is useful to note that C(n, k) can also be found using Pascal's triangle, as explained in the binomial coefficient article.