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Prove that sum(r=0)^n r(n-r)(^nC r)^2=n^...

Prove that `sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot`

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`underset(r=0)overset(n)sumr(n-r)(.^(n)C_(r))^(2) = underset(r=0)overset(n)sumr.^(n)C_(r)(n-r).^(n)C_(n-r)`
`=underset(r=0)overset(n)sumn.^(n-)C_(r-1)n.^(n-1)C_(n-r-1)`
`= n^(2)underset(r=0)overset(n)sum.^(n-1)C_(r-1).^(n-1)C_(n-r-1)`
`= n^(2) xx "coefficient of" x^(n-2) "in" (1+x)^(n-1)(1+x)^(n-1)`
`= n^(2) xx .^(2n-2)C_(n-2) = n^(2).^(2n-2)C_(n)`
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