I know only one nontrivial solution of this equation: $6!\cdot 7!=10!$. There is also a series of trivial solutions: $n!(n!-1)!=(n!)!,\ \forall n\in\mathbb{N}$. So my question is how to find any other solutions of the equation
$$a!b!=c!,\ a, b, c\in\mathbb{N}\quad(1)$$
I'm also interested in the solutions of more complex equations:
$$\prod\limits_{a\in A}a!=b!,\ A\subset\mathbb{N}\quad(2)$$
$$\prod\limits_{a\in A}a!=\prod\limits_{b\in B}b!,\ A,B\subset\mathbb{N},\ A\cap B=\varnothing\quad(3)$$
Update: as it is noted in comments both (1) and (2) equations have already appeared here. So the main question now is whether anything is known about equation (3).