Permutation and Combination Calculator

A permutation counts arrangements where order matters, so ABC and CBA are different. A combination counts selections where order does not matter, so ABC and CBA are the same group. For the same n and r, there are always at least as many permutations as combinations.

Order matters when the positions are distinct, such as first, second and third place in a race, a ranked podium, or a sequence like a PIN. Use combinations when you only care which items are picked, such as the players on a team or the numbers on a lottery ticket.

Permutations use nPr = n! / (n - r)!. Combinations use nCr = n! / (r! (n - r)!), which is the permutation count divided by r! to remove the duplicate orderings. The exclamation mark means factorial, the product of all whole numbers up to that value.

In a 6-from-49 lottery, order does not matter, so the number of possible tickets is the combination 49C6, which is 13,983,816. Picking a 5-player starting team from a squad of 12 is also a combination, 12C5 = 792. Choosing a captain, then a vice-captain from 12 is a permutation, 12P2 = 132, because the two roles are different.

Choosing zero items gives exactly one outcome (the empty selection), so nP0 and nC0 are both 1. Choosing all n items gives nCn = 1 because there is only one full set, while nPn = n! because every ordering of the full set is distinct.

Factorials grow extremely fast and quickly exceed the largest number a browser can store exactly, so n! is shown as too large to display once it overflows. The permutation and combination results are still computed step by step and stay accurate for moderate values of n even when n! itself cannot be shown.