Combination - Number of Combinations With Repetition

Number of Combinations With Repetition

A k-combination with repetitions, or k-multicombination, or multiset of size k from a set S is given by a sequence of k not necessarily distinct elements of S, where order is not taken into account: two sequences of which one can be obtained from the other by permuting the terms define the same multiset. In other words, the number of ways to sample k elements from a set of n elements allowing for duplicates (i.e., with replacement) but disregarding different orderings (e.g. {2,1,2} = {1,2,2}). If S has n elements, the number of such k-multicombinations is also given by a binomial coefficient, namely by

(the case where both n and k are zero is special; the correct value 1 (for the empty 0-multicombination) is given by left hand side, but not by the right hand side ). This follows from a clever representation of such combinations with just two symbols (see Stars and bars (combinatorics)).