Permutation & Combination Calculator
Compute permutations (nPr) and combinations (nCr) for choosing r items from a set of n.
Ordered arrangements of r items chosen from n.
720
120
3,628,800
Formula
About this calculator
Permutations and combinations count the ways you can select items from a larger set. A permutation counts arrangements where order matters — picking 1st, 2nd and 3rd place from a race, for example. A combination counts selections where order does not matter — like choosing three lottery numbers or members of a committee.
Both are built from factorials. The permutation formula nPr = n! / (n − r)! counts ordered selections, while the combination formula nCr = n! / (r! × (n − r)!) divides out the r! ways each group can be ordered. Because nCr ignores order, it is always less than or equal to nPr. Both require 0 ≤ r ≤ n, and very large values of n can produce results too big to represent exactly.
Frequently asked questions
A permutation counts arrangements where the order of selection matters, while a combination counts groups where order does not matter. nCr is always smaller than or equal to nPr.
n! (n factorial) is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
You cannot choose more items than exist in the set. If r exceeds n the formulas are undefined, so the calculator requires 0 ≤ r ≤ n.
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