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Prove that sum(r=0)^(2n) r.(""^(2n)C(r))...

Prove that `sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2n.""^(4n-1)C_(2n-1)`.

Text Solution

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`S=underset(r=0)overset(2n)sumr.(.^(2n)C_(r))^(2)`
`= underset(r=0)overset(2n)sum(r..^(2n)C_(r))(.^(2n)C_(r))`
`= underset(r=0)overset(2n)sum(2n)^(2n-1)C_(r-1)..^(2n)C_(2n-r)`
`= 2n`(Coefficient of `x^(2n-1)` in the expansion of `(1+x)^(2n-1)(1+x)^(2n))`
`= 2n`(coefficient of `x^(2n-1)` in the expansion of `(1+x)^(4n-1)`)
`= 2n xx .^(4n-1)C_(2n-1)`
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