Tuesday, 29 October 2013

combinatorics - Combinatorial identity using combinatorial argument: left[sumlimitsnk=0binomnkright]2=sumlimits2nk=0binom2nk



Is it possible to prove the following identity using combinatorial argument :




[nk=0(nk)]2=2nk=0(2nk)


Answer



Suppose that you have a group of n mixed couples. The lefthand side is the number of ways to choose a set of k men and a set of women for some k,{0,,n}; the righthand side is the number of ways to choose an arbitrary subset of the group. Clearly these are the same.


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