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*