What is a factorial (n!)

The factorial of a number, written n! ("n factorial"), is the product of all positive integers from 1 to n. It's a cornerstone of combinatorics.

How to calculate

n! = n × (n−1) × (n−2) × … × 2 × 1

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120. And since 4! = 24, you can reuse: 5! = 5 × 4!.

Why 0! = 1

It looks odd, but there's exactly one way to arrange "nothing" (the empty list), so 0! = 1 by definition. It also keeps combination and permutation formulas consistent.

Where factorials appear

• Permutations: how many ways to order n distinct items? Exactly n!. Ordering 5 books: 5! = 120.
• Combinations: C(n,k) uses factorials to count groups regardless of order.
• Probability and math series (like the one for e).

It grows staggeringly fast

Factorials explode: 10! already exceeds 3.6 million and 20! passes 2 quintillion.

FAQ

Is there a factorial of a negative or decimal? Not in the basic sense. For those, math uses the gamma function, a generalization of the factorial.