Permutations and Combinations

Permutations and combinations are fundamental counting techniques in discrete mathematics and probability. They answer two different questions: how many ways can I arrange items? And how many ways can I select items?

Permutations (Order Matters)

A permutation is an ordered arrangement. The number of permutations of n items taken r at a time is nPr = n! / (n-r)!. For example, the number of ways to arrange 3 books on a shelf from a collection of 10 is 10P3 = 720.

Combinations (Order Does Not Matter)

A combination is an unordered selection. The number of combinations of n items taken r at a time is nCr = n! / (r!(n-r)!). For example, the number of ways to choose a 3-person committee from 10 people is 10C3 = 120.

Factorial

The factorial n! = n × (n-1) × ... × 2 × 1. By convention, 0! = 1. Factorials grow extremely fast: 10! = 3,628,800 and 20! has 19 digits.

Practice with our Permutations & Combinations Calculator to verify your counting.