What Is a Factorial?
Definition
Factorial: The factorial of a natural number is the product of every whole number from 1 to that number, written n!: 5! = 120. By definition, 0! = 1.
A factorial is the product of every whole number from 1 to n, written n!: 5! = 1 × 2 × 3 × 4 × 5 = 120. It counts the ways to put n items in order. Combination and permutation formulas are quotients of factorials. Factorials grow very fast: 20! has 19 digits and 52! has 68.
Where it appears in the formulas
n! = 1 × 2 × … × n and 0! = 1. The combinations calculator shows each factorial of the formula as its own step: C(10, 3) = 10! / (3! × 7!) = 3,628,800 / (6 × 5,040) = 120.
Why is 0! equal to 1?
There is exactly one way to arrange no items. With this definition the formulas also work at the ends: C(n, 0) = n! / (0! × n!) = 1.