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Contents

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.