Calculate the number of permutations (nPr), the number of ways to arrange r items chosen from a set of n distinct items, where order matters.
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nPr = n! ÷ (n − r)!, the number of ways to select and arrange r items from a set of n distinct items, where the order of selection matters. For example, arranging 3 runners in 1st, 2nd and 3rd place out of 10 competitors gives 10P3 = 10 × 9 × 8 = 720 possible orderings.
The key distinction is whether order matters: permutations count arrangements (1st-2nd-3rd place is different from 2nd-1st-3rd), while combinations count selections regardless of order (a committee of 3 people is the same group no matter what order they were chosen in). Our Combination Calculator handles the order-doesn't-matter case.
For n = 10, r = 3: nPr = n! ÷ (n−r)! = 10! ÷ 7!, which simplifies to just the first 3 terms of 10!: 10 × 9 × 8 = 720. So there are 720 different ways to award 1st, 2nd and 3rd place among 10 competitors.
Using this calculator when order doesn't actually matter for your situation, like picking a committee, that's a combination (nCr), not a permutation, and will overcount how many distinct outcomes there really are.
Trying to compute full factorials for large n by hand, this calculator computes nPr as a running product of only r terms rather than full factorials, so it stays accurate and fast even for large n without needing n! itself.
Entering r greater than n, you can't select and arrange more items than exist in the set, this calculator requires r to be less than or equal to n.
It means swapping the position of two selected items creates a genuinely different outcome, like assigning gold, silver and bronze medals, versus just picking any 3 people for a team where their order of selection is irrelevant.
Then nPr simply equals n! (n factorial), the number of ways to arrange all n items in some order.
No, you can't arrange more items than exist in the set, so r must be less than or equal to n.
Combination Calculator
Calculate the number of ways to choose r items from n, where order doesn't matter.
Binomial Calculator
Calculate binomial probability for exactly k successes, or k or fewer, across n trials.
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