[latex]n! = n \times (n-1) \times \ldots 2 \times 1[/latex]

and that 0! is defined as 1.  But is there some higher level mathematics in which this factorial idea is just a corollary as with most pre-graduate school mathematics.  Well, there is!  This is the epic Gamma function.  It is defined as

[latex]\Gamma( \alpha ) = \int_0^{\infty} y^{\alpha – 1} e^{-y} dy[/latex].

If [latex]\alpha[/latex] is a positive integer, then [latex]\Gamma( \alpha ) = (\alpha – 1)![/latex].

Here is the main event.  Suppose [latex]\alpha = 1[/latex]; because [latex]\alpha[/latex] is a positive integer, [latex]\Gamma( \alpha ) = 0![/latex]  But then, notice that

[latex]\Gamma( 1 ) = \int_0^{\infty} y^{ 1 – 1 } e^{-y} dy = \int_0^{\infty} e^{-y} dy = 1[/latex].

Hence, our conclusion is that 0! = 1.