Permutation and Combination Calculator

A permutation counts arrangements where order matters; a combination counts selections where order does not. Ranking three runners is a permutation; choosing three runners for a relay pool is a combination.

They are the standard symbols: nPr is the number of permutations of r items chosen from n, and nCr (also written C(n,r) or "n choose r") is the number of combinations. Here n is the total and r is how many you take.

nPr = n! ÷ (n−r)! and nCr = n! ÷ (r! × (n−r)!). Equivalently, nCr = nPr ÷ r!, since each unordered selection of r items can be ordered in r! ways.

Because permutations distinguish orderings that combinations lump together. Each single combination corresponds to r! permutations, so dividing by r! can only shrink the count (or leave it equal when r is 0 or 1).

Choosing 0 items gives exactly one outcome (the empty selection), so nP0 = nC0 = 1. Choosing all n as a set also gives nCn = 1, while nPn = n! because all n can be ordered.

No. You cannot choose more distinct items than exist, so the tool reports an error when r exceeds n. Both counts would be zero in that case.

No. This calculates selections without repetition, where each item is used at most once — the usual textbook case. Problems that allow repeats (like combinations with replacement) use different formulas.

Up to 100,000. The results use big-integer arithmetic, so they are exact rather than rounded, though very large n with mid-range r can produce numbers with thousands of digits.

Each entry in Pascal’s triangle is a combination: row n, position r is nCr. So the triangle is just the table of "n choose r" values, and it explains the identity nCr = (n−1)C(r−1) + (n−1)Cr.

Yes. Everything is calculated in your browser; the numbers you enter are never uploaded and never reach analytics.