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# Factorials in Pascals Triangle

Hi,

I asked this question of Keith Conrad, and he suggested that I try posting here. One of my students observed that the only instances of factorials in the interior of Pascal's triangle are $\binom{4}{2}=3!$ and $\binom{10}{3}=\binom{16}{2}=5!$. I checked the first 500 rows, and he's right up to that point.

This is a special case of the apparently unsolved problem of finding non-trivial solutions to n!=a!b!c!... The special feature here is that I need (a+b)!=a!b!c!, and that seems like a special enough case to have been treated by someone. Unfortunately, literature searches have been fruitless, because every paper about Pascal's triangle contains the word "factorial" somewhere.

My best idea (which I can't make work) is to show that the powers of 7 in the equation (a+b)!=a!b!c! can't be made to match up unless neither side is a multiple of 7. Then exhaustive searching can show that the above are the only non-trivial solutions.

Thanks greatly for any ideas that anyone might have.