factorial in nLab

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Definition

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

k!≔1⋅2⋅3⋅4⋅⋯⋅(k−1)⋅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=0k = 0 by setting 0!=10! = 1. This may be justified by defining k!k! to be the number of permutations of a set with kk elements.

• double factorial

• binomial coefficient, multinomial coefficient

• Gamma function

• Euler beta function

• exponential series

References

See also

• Wikipedia, Factorial