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 being a mathematician (amateur or otherwise), I only noticed a handful of these.
Of course there are a couple of trivial cases to negate: the case when $n$ arbitrary $m = n!$ and $j=1$, likewise when $m$ arbitrary and $j=0$ and $n=0,1$.
I decided to write a Mathematica code to check this for me which I can make available for anyone who is interested. I decided to compute up to $30!$, i.e. $n=30$, and $m$ up to $110$. There were only $3$ binomial coefficients that were factorials (that were not trivial) up to binomial coefficient symmetry:
- $\dbinom{4}{2} = 6 = 3!$
- $\dbinom{10}{3} = 120 = 5!$
- $\dbinom{16}{2} = 120 = 5!$
Given how early these occur in the computations, it suggests that there are only finitely many such binomial coefficients - neglecting the trivial cases. Is there anything known in this direction? Are these actually the only ones or are there more lurking out there?