Permutation and Combination Calculator
nPr = n! / (n - r)!. nCr = n! / (r! x (n - r)!).
Two numbers in, both answers out
Enter the total number of items you are choosing from, n, and how many you are choosing, r. The calculator returns both the permutation count and the combination count, so you do not have to commit to one before you see them.
Showing both is the whole idea. The same n and r produce very different totals depending on whether order counts, and seeing the two figures together makes that gap impossible to miss.
Combinations count groups, permutations count orderings
A combination is a group where the lineup does not matter. Picking Ana, Ben and Cara for a committee is the same combination however you list their names. A permutation treats each ordering as distinct, so seating those three people in first, second and third place counts as a different result each time the order changes.
That is why the permutation count is always the larger of the two, or equal when you are choosing everything. Every combination hides several permutations inside it, one for each way of ordering the same group.
The two formulas
Permutations use n factorial divided by the factorial of n minus r. That counts every ordered way to line up r items out of n.
Combinations take that same number and divide it once more by r factorial, which cancels out all the orderings of each group. In short, nCr is nPr with the order stripped back out.
Choosing three from five
Suppose five people and three spots. The number of combinations, 5C3, is 10: ten possible committees. The number of permutations, 5P3, is 60.
The factor between them is 3 factorial, which is 6, because each committee of three can be ordered six different ways. Ten groups times six orderings is sixty arrangements.
Deciding whether order matters
The whole choice comes down to one question about your problem. If rearranging the same items gives you a genuinely different outcome, you want permutations. If it does not, you want combinations.
A medal podium of gold, silver and bronze is a permutation, since swapping two athletes changes who won what. A hand of lottery numbers is a combination, since the order they are drawn in makes no difference to whether you match.
Where these counts get used
Combinations sit behind lottery odds, card hands and any committee or team selection. They answer how many distinct groups are possible.
Permutations sit behind passwords, race finishes, seating plans and lock codes, anywhere the sequence itself is part of the answer. Probability problems lean on both, usually as the denominator that counts every equally likely outcome.
Permutation and Combination Calculator FAQ
What is the difference between a permutation and a combination?
Permutations count arrangements where order matters. Combinations count groups where order does not.
For the same n and r, the permutation count is always at least as large as the combination count.
When do I use nCr?
Use combinations when order does not matter, such as choosing a committee, a set of pizza toppings or lottery numbers.
Only the group counts, not the sequence the items came in.
When do I use nPr?
Use permutations when order matters, such as finishing places in a race, the characters in a password or a seating arrangement.
Each distinct ordering is counted as its own outcome.
What does n choose r mean?
It is the spoken name for nCr, the number of ways to choose r items from n without regard to order.
Five choose three is 10.
Can r be larger than n?
No. You cannot choose more items than you have, so the calculator requires r to be no greater than n.