Permutation & Combination Calculator
Give it n and r and it hands back both nPr and nCr — the count when order matters, and the count when it doesn't.
How it works
Both start with n items and choose r of them. The difference is whether the order counts. Permutations (nPr) treat ABC and CBA as different arrangements; combinations (nCr) treat them as the same group.
Permutations use nPr = n! / (n − r)!, and combinations use nCr = n! / (r!(n − r)!). We compute them by multiplying out only the terms we need instead of full factorials, so big values don't blow up into overflow.
Picking 3 people from 10 for a race where finishing order matters gives 10P3 = 720 outcomes. Picking a committee of 3 from the same 10, where order is irrelevant, gives 10C3 = 120 — six times fewer, since each committee could be arranged 3! = 6 ways.
Frequently asked questions
When do I use a permutation versus a combination?
Ask whether order changes the outcome. Ranking finishers, assigning specific roles, or building a passcode — order matters, so use permutations. Choosing a team, a hand of cards, or a set of toppings — order is irrelevant, so use combinations.
Why is nPr always bigger than nCr?
Because permutations count every ordering of the same group as separate. Each combination of r items can be shuffled r! ways, so nPr is exactly nCr times r!. More orderings means a larger total.
What happens if r is bigger than n?
You get a dash, because you can't choose more items than you have. Pick r somewhere between 0 and n and both results appear. Whole numbers only — you can't select half an item.
How big can the numbers get?
Very. Factorial growth is steep, so even modest n and r produce huge counts. The calculator multiplies terms iteratively to stay accurate as long as possible, but extreme inputs can push past what a browser can represent exactly.