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Contents
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 nCk, nCk or as
- <math>{n \choose k},</math>
or occasionally as C(n, k).
One method of deriving a formula for nCk proceeds as follows:
- 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.
- 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.
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