Permutation and Combination Calculator

The calculation itself

nCr = n! ÷ (r! × (n − r)!) counts selections, where order does not matter. nPr = n! ÷ (n − r)! counts arrangements, where it does.

They differ by exactly r!, because every selection of r items can itself be ordered in r! ways. That single factor is the whole distinction between the two.

In practice

Five cards from a standard deck: 2,598,960 combinations and 311,875,200 permutations. The ratio is 120, which is 5! — the number of orders a single hand can be dealt in.

Six numbers from 49: 13,983,816 combinations. That is the odds of a lottery jackpot, and stating it as one in fourteen million is more honest than any smaller-sounding rearrangement of the same figure.

With repetition allowed the numbers change shape entirely. Three dice give 6³ = 216 ordered outcomes; two scoops from four ice cream flavours, where a double scoop of one flavour counts, give 10 rather than 6.

Why the naive formula fails

Computing 52C5 from three factorials means evaluating 52!, a number with 68 digits. It overflows ordinary floating point long before the division brings it back down.

The answer, 2,598,960, is small. So this computes it multiplicatively — multiplying and dividing term by term so the running value never grows large — and uses the smaller of r and n−r, since nCr equals nC(n−r).

It is a small implementation detail with a real consequence: the naive method fails on precisely the deck-of-cards and lottery cases that people come to a combinations calculator to work out.