Factorial

discrete-math

Definition

Factorial

The factorial of a natural number $n\in \mathbb{N}$ is the product of all positive natural numbers less than or equal to $n$:

$$n!:=\prod _{k=1}^{n}k=\{\begin{array}{ll}1\cdot 2\cdot \cdots \cdot n& \text{if}n\ge 1\\ 1& \text{if}n=0\end{array}$$

The factorial function can also be defined recursively:

$$n!=\{\begin{array}{ll}1& \text{if}n=0\\ n(n-1)!& \text{if}n>0\end{array}$$