Permutations Calculator

Permutations are arrangements of items where order matters. ABC, ACB, BAC, BCA, CAB, and CBA are six different permutations of the same three letters. The standard P(n, r) counts how many ways you can arrange r items chosen from a set of n distinct items.

Without repetition, P(n, r) = n! ÷ (n − r)!. With repetition, the count is simply n raised to the r power (n^r), because each of the r positions can be any of the n items independently. The calculator offers both modes via the toggle.

Without repetition is the right mode when an item can only appear once in the arrangement: who finishes 1st, 2nd, 3rd in a race, or which 4 letters spell a word using each letter only once. With repetition is the right mode when items can repeat: 4-digit PIN codes where digits can repeat, or license plates where any letter can appear in any slot.

Permutations count arrangements where order matters; combinations count selections where order does not. Choosing the gold, silver, and bronze winners is a permutation. Choosing 3 finalists out of a pool is a combination. P(n, r) is always at least as large as C(n, r), and equals r! × C(n, r).

P(n, 0) = 1 by convention: there is exactly one way to arrange zero items (the empty arrangement). Similarly, 0! = 1 by convention, which makes the formula P(n, n) = n! ÷ 0! = n! consistent.

In without-repetition mode, that is impossible: you cannot pick more distinct items than the set contains, and the calculator flags it. In with-repetition mode, r can freely exceed n because each position is filled independently. A 6-character password from a 4-letter alphabet has r > n and is perfectly valid (4^6 = 4,096 possibilities).