Why is 0 factorial equal to 1? Is there any pure basic mathematical proof?

Question

I just got a question while reading permutation. Why 0 is factorial equal to 1?

Answer 1

One definition of the factorial that is more general than the usual

$$ N! = N\cdot(N-1) \dots 1 $$

is via the gamma function, where

$$ \Gamma(N) = (N-1)! = \int_0^{\infty} x^{N-1}e^{-x} dx $$

This definition is not limited to positive integers, and in fact can be taken as the definition of the factorial for non-integers. With this definition, you can quite clearly see that

$$ 0! = \Gamma(1) = \int_0^{\infty} e^{-x} dx = 1 $$

If you are starting from the "usual" definition of the factorial, in my opinion it is best to take the statement $0! = 1$ as a part of the definition of the factorial function, as anything else would require proofs using the factorial to include special cases for $0!$ and $1!$. It's a definition that is consistent and makes our lives easier.

Answer 2

There is exactly one permutation of a set of zero elements.