kpermutations

In combinatorics, a k-permutation of a set of n distinct elements is an ordered arrangement of k distinct elements chosen from the set. The concept is used when the order of selection matters and repetitions are not allowed. It is denoted by P(n, k) or _nP_k, and is defined for integers 0 ≤ k ≤ n. When k > n, the number of k-permutations is typically considered undefined.

Number of k-permutations without repetition: P(n, k) = n! / (n - k)!. This can be understood as multiplying the choices available for each position: n × (n-1) × … × (n-k+1). For example, from a set of five items {A,B,C,D,E}, the number of 3-permutations is 5 × 4 × 3 = 60, which equals 5! / (5-3)! = 120/2 = 60. Some simple special cases: P(n,0) = 1 and P(n,n) = n!.

Variant with repetition: If repetition of elements is allowed and order matters, the number of length-k sequences is n^k. If the problem instead asks for unordered selections with repetition allowed, that is combinations with repetition, which are given by binomial coefficients for multisets.