Permutation and Combination Calculator

How many ways to arrange (nPr) or choose (nCr) r items from a group of n.

Permutations count order as significant; combinations don't.

Permutations (order matters) — nPr
20
Combinations (order doesn't matter) — nCr
10

Picking a 1st, 2nd and 3rd place winner from a group is a permutation (order matters). Picking any 3 people for a committee is a combination (order doesn't matter).

A worked example

Choosing 6 numbers from 49 for a lottery (where order doesn't matter) gives 13,983,816 possible combinations — the famous odds behind a standard 6/49 lottery game.

Frequently asked questions

How do I know if my problem needs a permutation or a combination?

Ask whether order matters. Assigning 1st, 2nd, and 3rd place is a permutation — switching who gets which place changes the outcome. Picking any 3 people for a team is a combination — the order you pick them in doesn't matter, just who ends up on the team.

Why is nPr always bigger than nCr for the same n and r (when r > 1)?

Permutations count every possible ordering of the same r items as a separate outcome, while combinations group all those orderings together as one — nPr is always nCr multiplied by r! (the number of ways to order r items).