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### When is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $n,m,j$?

As stated in the title: when is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $m,j,n$? I was thinking about this problem a couple of days ago because in all my years of ...
Let $F = \{n! \mid n > 1\}$. Let $S_1$ be the set of integers which can be expressed as a product of one or more (not necessarily distinct) elements of $F$. Let $S_2$ be the set of integers which ...