Factorial is the function, denoted n!, that is the product of the positive integers less than or equal to n. For example, 1! = 1; 5! = 5 × 4 × 3 × 2 × 1 = 120; 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800. 0! is defined to be 1, by working the relationship n! = n × (n–1)! backward.

 

An interesting equality is 1! 10! 22! 1! = 11! 0! 2! 21! in which the same digits are broken up two different ways into factorials. This may be the smallest such example.

 

Factorials are important in combinatorics because there are n! different ways (permutations) of arranging n distinct objects in a sequence. They also turn up in formulas in calculus, for instance in Taylor's theorem, because the n-th derivative of the function x ⁿ is n!.

 

For large numbers the approximation n! = n ⁿ.e ^–n√(2πn) can be used. This is called Stirling's formula.