Sunday, 8 May 2016

combinatorics - Combinatorial proof of nchoosepnchooseq=sumnk=0nchooseknkchoosepknkchooseqk

I am trying to do a combinatorial proof of {n\choose p}{n\choose q}=\sum_{k=0}^{n} {n \choose k}{n-k \choose p-k}{n-k \choose q-k}



For the left side. I thought of two urns with n red and n blue balls and choosing p-red balls and q-blue balls.



For the right side, i am not very sure, but I thought of make k the number of couples of red and blue balls. Making this is {n \choose k} ways. Since it's the same counting {n \choose k} or {n \choose n-k}. I choose {n-k \choose p-k} red balls and the same way with blue.




But I do not think this is right, any help will be appreciated.

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