Combination
From Freepedia
In combinatorial mathematics, a combination of members of a set is a subset. 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 or k-subsets 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.
Efficient calculation
In trying to find the value of (for example) <math>{20 \choose 4}={20! \over 4! 16!},</math> one should not compute 20!, which is huge number. Instead, an initial cancellation of 20! with 16! yields
- <math>{20 \cdot 19 \cdot 18 \cdot 17 \over 4 \cdot 3 \cdot 2 \cdot 1}</math>
and then do further cancellations before multiplying. The denominator in every case cancels out completely.
