Permutations and Combinations

The number of r-permutations of a set of n objects with repetition allowed is nrn^r.

There are C(n+r−1,r)C(n + r - 1, r) r-combinations from a set of n elements when repetition of elements is allowed.

The number of permutations of n objects, where there are n1n_1 indistinguishable objects of type 1, n2n_2 indistinguishable objects of type 2, …, and nkn_k objects of type k, is

n!n1!n2!…nk!\frac{n!}{n_1!n_2!\dots n_k!}

Type Repetition Allowed? Formula
r-permutations no n!(n−r)!\frac{n!}{(n-r)!}
r-combinations no n!r!(n−r)!\frac{n!}{r!(n-r)!}
r-permutations yes nrn^r
r-combinations yes (n+r−1)!r!(n−r)!\frac{(n+r-1)!}{r!(n-r)!}

References