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# factorial

For any non-negative integer $n$, the factorial of $n$, denoted $n!$, can be defined by

$n!=\prod_{{r=1}}^{n}r$ |

where for $n=0$ the empty product is taken to be $1$.

Alternatively, the factorial can be defined recursively by $0!=1$ and $n!=n(n-1)!$ for $n>0$.

$n!$ is equal to the number of permutations of $n$ distinct objects. For example, there are $5!$ ways to arrange the five letters A, B, C, D and E into a word.

For every non-negative integer $n$ we have

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

where $\Gamma$ is Euler’s gamma function. In this way the notion of factorial can be generalized to all complex values except the negative integers.

Related:

BinomialCoefficient, ExponentialFactorial

Synonym:

factorial function

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

05A10*no label found*11B65

*no label found*92-01

*no label found*92B05

*no label found*

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