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Give a combinatorial proof that $\displaystyle{}\sum_{k = 1}^{n} k \binom{n}{k}^{2} = n \binom{2n−1}{n−1}.$ [Hint: Count in two ways the number of ways to select a committee, with $n$ members from a group of $n$ mathematics professors and $n$ computer science professors, such that the chairperson of the committee is a mathematics professor.]
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