What is 0!

0! equals 1;

One of the ways to show this is to "complete the pattern":

$$\\5!=1\times 2\times 3\times 4\times 5=120 \\4!=\frac{5!}{5}=\frac{120}{5}=24 \\3!=\frac{4!}{4}=\frac{24}{4}=6 \\2!=\frac{3!}{3}=\frac{6}{3}=2 \\1!=\frac{2!}{2}=\frac{2}{2}=1 \\0!=\frac{1!}{1}=\frac{1}{1}=1$$

But that's where the pattern ends; you can't continue, because you end up with this:

$$(-1)!=\frac{0!}{0}=\textcolor[rgb]{1,0,0}{\frac{1}{0}}$$

See what I mean ?

$${\mathtt{0}}{!} = {\mathtt{1}}$$