The Factorial Function

# The Factorial Function

Definition: The Factorial Function $f: \{0, 1, 2, ... \} \to \{0, 1, 2, ... \}$ is defined by $f(n) = n \cdot (n - 1) \cdot ... \cdot 2 \cdot 1 = \prod_{i=1}^{n} i = n!$. |

By convention we define $0! = 1$. The table below gives $n!$ for the first few values of $n$.

$0! = 1$ | $11! = 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 39916800$ |

$1! = 1$ | $12! = \prod_{i=1}^{12} i =479001600$ |

$2! = 2 \cdot 1 = 2$ | $13! = \prod_{i=1}^{13} i = 6227020800$ |

$3! = 3 \cdot 2 \cdot 1 = 6$ | $14! = \prod_{i=1}^{14} i = 87178291200$ |

$4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$ | $15! = \prod_{i=1}^{15} i = 1307674368000$ |

$5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ | $16! = \prod_{i=1}^{16} i = 20922789890000$ |

$6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$ | $17! = \prod_{i=1}^{17} i = 355687428100000$ |

$7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5040$ | $18! = \prod_{i=1}^{18} i = 6402373706000000$ |

$8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 40320$ | |

$9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 362880$ | |

$10! = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 3628800$ |

As you can see, the factorial function grows extraordinarily fast. Computing $50!$ would be an astronomically large number, let alone $100000!$. This extreme growth rate is sometimes referred to as an example of a *combinatorial explosion*.