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.

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