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Give a combinatorial proof that if n is a positive integer then $\displaystyle\sum_{k = 0}^{n} k^{2} \binom{n} {k} = n(n + 1)2^{n−2}.$ [Hint: Show that both sides count the ways to select a subset of a set of $n$ elements together with two not necessarily distinct elements from this subset. Furthermore, express the right-hand side as $n(n − 1)2^{n−2} + n2^{n−1}.]$