Give a combinatorial proof that if n is a positive integer
Chapter 9, Problem 38E(choose chapter or problem)
Give a combinatorial proof that if \(\mathrm{n}\) is a positive integer then \(\sum_{k=0}^{n} k^{2}\left(\begin{array}{l}n \\ k\end{array}\right)=n(n+1) 2^{n-2}\). [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 \(\left.n(n-1) 2^{n-2}+n 2^{n-1} .\right]\).
Equation Transcription:
Text Transcription:
sum_k=0^n k^2(_k^n)=n(n+1)2^n-2
n(n-1)2^n-2+n2^n-1
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