Factorial Symmetry

There are already not a lot of numbers who have the same product as they do sum. Let's first find these numbers, then restrict them only to factorials. Our equation is

$AB=A+B$

Solving for the variable $B$ , we have

$B=\dfrac{A+B}{A}$

$B=\dfrac{B}{A}+1$

We can conclude that $B \geq A$ . But solving for $A$ , we get

$A=\dfrac{A}{B}+1$

Which means $A \geq B$ . For both these conditions to be true, $A=B$ . Now we have a one variable quadratic:

$A^2=2A$

Which clearly has roots at $A=0, 2$ . Only $\boxed{1}$ of these solutions is a perfect factorial, $2$ , since $2!=2$ .