Factorial

For alternative uses of the exclamation point symbol, please see double factorial for expressions
of the form n!! or subfactorial for expressions of the form !n .

Factorial is a function denoted by a trailing exclamation point (!), which is defined for all non-negative integers.

For any positive integer, it outputs the product of all natural numbers between 1 and that number, inclusive:

$n!\equiv1\cdot2\cdot3\cdots n$

The notation $n!$ is read "$n$ factorial". Alternatively, one could think of the product as being in the opposite order:

$n!\equiv n(n-1)(n-2)\cdots3\cdot2\cdot1$

As a concrete example:

$5!=1\cdot2\cdot3\cdot4\cdot5=120$

As a consequence of the empty product,

$0!\equiv1$

You can further prove this by saying that $n!={\frac {(n+1)!}{n+1}}$. For example,

$4!={\frac {(4+1)!}{4+1}}={\frac {5!}{5}}={\frac {120}{5}}=24$

Therefore,

$0!={\frac {(0+1)!}{0+1}}={\frac {1!}{1}}={\frac {1}{1}}=1$

Factorials are commonly used in combinatorics and probability theory. It is also used in Taylor polynomials and infinite series.

The factorial function can also be seen as a specific case of the gamma function ($\Gamma$), which extends the factorial to the complex plane (excluding the non-positive integers). In particular, for all values for which the factorial is defined:

$n!=\Gamma(n+1)$

Examples

$n$	$n!$
0	1
1	1
2	2
3	6
4	24
5	120

External links

• Fast Factorial Functions: Contains algorithms for computing large factorials.