Finding natural numbers $a, b, c$ such that $a!=b!\times c!$

Question

Yesterday, when I was playing with numbers, I was surprised to know the following relation:

$$10!=6!\times 7!.$$

Then, I've been looking for the other solutions, but I'm facing difficulty. Then, here are my questions.

Question 1 : Does there exist the other non-trivial solution?

Question 2 : Can we get all solutions? In addition to this, can we prove that they are all solutions?

I don't know if these questions are famous. Also, I'm afraid that these questions might be very easy to solve. Anyway, I need your help.

Answer 1

Hint: Notice that $7$ is the largest prime $\leq 10$.