Permutations and Combinations Calculator

This permutations and combinations calculator works out how many ways you can choose and arrange items from a larger set. You enter the total number of items, n, and the number you are selecting, r, and the tool returns the number of permutations, written nPr, and the number of combinations, written nCr. A permutation counts ordered arrangements, so the order in which you pick items matters, while a combination counts unordered selections, where only which items you pick matters and not their order. The permutation formula is n factorial divided by n minus r factorial. The combination formula divides that result by r factorial, because each unordered group can be arranged in r factorial different orders. Students, teachers, lottery and probability enthusiasts, and anyone working with counting problems use these figures to count outcomes, work out probabilities and plan experiments. Three tips help you use the tool well. First, decide whether order matters before you start, since ordered tasks like ranking finishers or setting a passcode use permutations, while unordered tasks like choosing a committee or a hand of cards use combinations. Second, nPr is always at least as large as nCr for the same n and r, because every combination produces r factorial permutations, so if your answer for combinations is larger than permutations you have swapped the two. Third, r cannot be greater than n, so check your inputs if you get a zero or an error. These counts grow extremely fast, so even modest values of n can produce very large numbers.

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720
Permutations (nPr)
Combinations (nCr)120
r factorial6

nPr = n! / (n-r)!. nCr = n! / (r! (n-r)!).

How it works

Permutations are n factorial divided by n minus r factorial, counting ordered arrangements. Combinations divide that by r factorial to remove the order, counting unordered selections. If r is greater than n both results are zero.

Worked example

With n of 10 and r of 3, permutations are 10 times 9 times 8, which is 720. Dividing by 3 factorial, which is 6, gives 120 combinations.

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Frequently asked questions

What is the difference between a permutation and a combination?

A permutation counts ordered arrangements, so picking A then B is different from B then A. A combination counts unordered selections, so A and B is the same group either way. Use permutations when order matters and combinations when it does not.

Why is nPr always larger than nCr?

Every combination of r items can be arranged in r factorial different orders, and each of those counts as a separate permutation. So the number of permutations is the number of combinations times r factorial. That makes nPr at least as large as nCr for the same n and r.

What happens if r is bigger than n?

You cannot choose more items than you have, so both nPr and nCr are zero. The calculator treats r greater than n as having no valid arrangements. Make sure r is between zero and n for a meaningful result.