For alternative uses of the exclamation point symbol, please see double factorial for expressions
of the form 4!! or subfactorial for expressions of the form !3 .


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:

${\displaystyle n!\equiv1\cdot2\cdot3\cdots n}$

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

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

As a consequence of the empty product,

${\displaystyle 0!\equiv1}$

As a concrete example:

${\displaystyle 5!=1\cdot2\cdot3\cdot4\cdot5=120}$

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 (${\displaystyle \Gamma}$), which extends the factorial to the complex plane (excluding the non-positive integers). In particular, for all values for which the factorial is defined:

${\displaystyle n!=\Gamma(n+1)}$