The factorial of a positive integer, *n*, denoted *n*!, is the product of the positive integers less than or equal to *n*. E.g.,
*n* is a positive integer.
By using this relation, we can extend the definition of factorials and define *z*! for all complex numbers *z* except the negative integers.

- 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)! backwards.

Sometimes, *n*! is read 'n shriek', in reference to the exclamation mark notation.

A good approximate formula for factorials is *n*! ~ (2 π *n*)^{1/2} (*n*/e)^{n}, which is known as **Stirling's Formula**, after [James Stirling]?, the mathematician who discovered it. It is quite accurate when *n* is large, however it has to be interpreted right: it means that the *quotient* of the two functions approaches 1 as *n* approaches infinity; it does not mean that their *difference* approaches zero.

The related gamma function Γ(*z*) can be defined for all complex numbers *z* except for *z* = 0, -1, -2, -3, ... It has the property

- Γ(
*n*+1) =*n*!

Factorials are important in combinatorics because there are *n*! different ways of arranging *n* distinct objects in a sequence (see permutations). 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*!.