nLab factorial

Definition

For $k \in \mathbb{N}$ a natural number, its factorial $k! \in \mathbb{N}$ is the number obtained by multiplying all positive natural numbers less than or equal to $k$:

$$k! \;\coloneqq\; 1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot (k-1) \cdot k \,.$$

In combinatorics, the definition usually extends to $k = 0$ by setting $0! = 1$. This may be justified by defining $k!$ to be the number of permutations of a set with $k$ elements.

Related concepts

• double factorial

• binomial coefficient, multinomial coefficient

• Gamma function

• Euler beta function

• exponential series

References

See also

• Wikipedia, Factorial