Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?

Answer

We need $0!$ to be defined as $1$ so that many mathematical formulae work. For example we would like
$$n! = n \times (n-1)!$$
to work when $n=1,$ ie $1! = 1 \times 0!.$
Also we require that the formula for the number of ways of choosing $k$ objects from $n$ is valid for $k=n.$ ie
$${n \choose k} = \frac{n!}{k!(n-k)!}$$
is valid when $k=n.$

Things need to work when we extend our definition of the factorial via the gamma function.

$$\Gamma(z) = \int\limits_0^\infty t^{z-1} e^{-t} \,\mathrm{d}t,\qquad \Re(z)>0.$$

The above gives $\Gamma(n)=(n-1)!$ and so we require $0!=1,$ since $\Gamma(1)=1.$

Attribution
Source : Link , Question Author : Ssegawa Victor , Answer Author : Sasha

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