Permutations P(n,k) Calculator

Permutations P(n,k): formula and examples

A permutation (also called an arrangement) of n distinct objects taken k at a time, where order matters, comes from P(n,k) = n! / (n−k)!. Say you want to hand out gold, silver and bronze among 8 runners. That's P(8,3) = 8·7·6 = 336 different podiums. Set k = n and you fall back to the total permutation P(n) = n!, which counts every ordering of all n elements. What sets this apart from combinations is order. With P(n,k) the sequence ABC and the sequence CBA each count once; with C(n,k) they're treated as the same thing.

Applications: rankings, PINs and shuffling

Permutations show up wherever order carries meaning: the way a queue lines up, who lands on each step of a podium, electing officers to distinct roles (chair, secretary, treasurer), PIN codes with no repeated digits (P(10,4)=5040), license plates, and the Fisher-Yates shuffle behind audio players and card games.

FAQ

Permutation vs combination? Ask yourself whether shuffling the items changes anything. If it does (podium, password, sequence), reach for permutation. If it doesn't (committee, lottery), use combination.

What if k = 0? P(n,0) = 1. There's a single way to arrange zero objects, namely the empty arrangement.

What if k > n? P(n,k) = 0. You can't pick out more distinct objects than you actually have.

Are repetitions allowed? No. P(n,k) takes the elements to be distinct. When repetition is on the table, use n^k instead.