I am trying to do a combinatorial proof of (np)(nq)=∑nk=0(nk)(n−kp−k)(n−kq−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 (nk) ways. Since it's the same counting (nk) or (nn−k). I choose (n−kp−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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