nPr

In combinatorics, nPr denotes the number of r-permutations of n distinct items. It counts the number of ways to choose and arrange r items from a set of n, with order mattering and no item used more than once.

The standard formula is nPr = n! / (n − r)!, valid for integers n ≥ r ≥ 0. Special cases include nP0 = 1, and nPn = n!. If r > n, most conventions set nPr = 0 or undefined.

Example: 5P3 = 5 × 4 × 3 = 60.

nPr is related to combinations by the identity nPr = nCr × r!, since one can first choose r items (nCr ways) and then arrange them (r! ways).

Repetition context: nPr refers to selections without repetition. If repetition is allowed, the number of r-permutations is n^r, which is a different counting problem.

Applications of nPr include counting possible lineups, seating arrangements, distinct password or code orders, and any scenario where the order of selected items matters and no item is reused.