Sunday, 8 May 2016

combinatorics - Combinatorial proof of ${nchoose p}{nchoose q}=sum_{k=0}^n {n choose k}{n-k choose p-k}{n-k choose q-k}$

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.

No comments:

Post a Comment

real analysis - How to find $lim_{hrightarrow 0}frac{sin(ha)}{h}$

How to find $\lim_{h\rightarrow 0}\frac{\sin(ha)}{h}$ without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...