Permutation and Combination Calculator

Enter n and r to calculate permutations (nPr) and combinations (nCr) together.

How to Use the Permutation and Combination Calculator

Enter n for the total number of items and r for how many get chosen — both nPr and nCr calculate together, since seeing them side by side is the point.

Order is the entire distinction between these two numbers. Permutations count arrangements where order matters: first, second, and third place in a race are three different outcomes, and swapping two runners changes the result even if the same three people finish. Combinations count selections where order doesn't matter — pick a 3-person committee out of 10 people, and the same three people chosen in any order are still the same committee.

nPr = n! ÷ (n − r)!
nCr = n! ÷ (r! × (n − r)!)

With the calculator's own defaults, n = 10 and r = 3 gives 720 permutations against just 120 combinations — the same ten people, the same three chosen, but six times as many ways to order them into 1st, 2nd, and 3rd. nPr is never smaller than nCr for the same n and r, and the gap between them is exactly r! — every unique combination corresponds to r! different orderings of the same items, so permutations always "overcount" by that factor. The calculator surfaces the ratio directly: divide nPr by nCr and you land back on r!, a clean relationship worth checking by hand once.

Factorials grow fast — 170! is close to the largest value a browser's floating-point math can represent before it overflows to infinity — so n above 170, or an r larger than n, returns a plain error instead of a silently wrong number.