# factorial

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*√(2

^{ –n}*πn*) can be used. This is called

**Stirling's formula**.