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