Definition:Factorial

Definition
Let $n \in \N$.

Then the factorial of $n$ is defined inductively as:
 * $n! = \begin{cases}

1 & : n = 0 \\ n \left({n - 1}\right)! & : n > 0 \end{cases}$

That is:
 * $n! = \displaystyle \prod_{k \mathop = 1}^n k = 1 \times 2 \times \cdots \times \left({n-1}\right) \times n$

where $\prod$ denotes product notation.

The first few factorials are:

$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \ 040 \\ 8 & 40 \ 320 \\ 9 & 362 \ 880 \\ 10 & 3 \ 628 \ 800 \\ \end{array}$

...etc.

Also see

 * Definition:Gamma Function: an extension of the concept of the factorial to the complex plane.