# 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

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

where $\mathrm{\Gamma}$ is Euler’s gamma function^{}.
In this way the notion of factorial can be generalized to all complex (http://planetmath.org/Complex) values except the negative integers.

Title | factorial |

Canonical name | Factorial |

Date of creation | 2013-03-22 11:53:58 |

Last modified on | 2013-03-22 11:53:58 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 22 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 05A10 |

Classification | msc 11B65 |

Classification | msc 92-01 |

Classification | msc 92B05 |

Synonym | factorial function |

Related topic | BinomialCoefficient |

Related topic | ExponentialFactorial |