Thanks, my machine is the same as yours and runs out of memory too.
Coming at the problem in another way, I get for 5 pairs out of 13 players:
(1/3840) (m - 9) (m - 8) (m - 7) (m - 6) (-5 + m) (-4 + m) (-3 + m) (-2 + m) (-1 + m) m /. m -> 13
I have some confidence in the result now. I would like to have a solution in generating functions too but so far this has not been possible.
I am glad you asked me that question. EM mean Experimental Mathematics.
I would say the first rule is this:
Get Your Hands Dirty: This is easy and fun to do. Stay loose and experiment. Plug in lots of numbers. Keep playing around until you see a pattern. Then play around some more, and try to figure out why the pattern you see is happening. It is a well-kept secret that much high-level mathematical research is the result of low-tech "plug and chug" methods. The great Gauss, widely regarded as one of the greatest mathematicians in history, was a big fan of this method. In one investigation, he painstakingly computed the number of integer solutions to x^2 + y^2 < 90,000
Today, we can be assisted by a computer to help with this type of research.