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
_{n}C
_{k},
^{n}C
_{k} or as
- <math>{n \choose k},</math>
or occasionally as C(
n,
k).
One method of deriving a formula for _{n}C_{k} 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.
All Wikipedia text
is available under the
terms of the GNU Free Documentation License