User rob gross - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:55:12Z http://mathoverflow.net/feeds/user/4377 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17058/factorials-in-pascals-triangle Factorials in Pascals Triangle Rob Gross 2010-03-04T05:30:34Z 2010-10-04T13:28:50Z <p>Hi,</p> <p>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.</p> <p>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.</p> <p>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. </p> <p>Thanks greatly for any ideas that anyone might have.</p>