Topic
Factorials
The factorial of a nonnegative integer n is the product of the positive integers from 1 through n, written with an exclamation mark such as 5!.
At a glance
- Notation
- n!
- Special value
- 0! = 1
Overview
Factorials grow very quickly and appear in counting, probability, permutations, combinations, power series, and other areas of mathematics. By convention, zero factorial equals one. The notation n! is defined for nonnegative integers in elementary combinatorics, while the gamma function extends related behavior beyond those integers.
A small example
Five factorial is 5 × 4 × 3 × 2 × 1 = 120. The rapid growth means even moderately sized factorials become enormous, which is why software often needs arbitrary-precision arithmetic or logarithmic methods instead of ordinary fixed-width integers.
Why zero factorial equals one
The convention 0! = 1 keeps counting formulas consistent, including the number of ways to arrange zero objects and identities involving combinations. It also preserves the recurrence n! = n × (n-1)! when moving from 1! down to 0!.
Sources and review
MOOR's explanatory text is supported by the following source links.