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Question

The value of
12.nC1+22.nC2+32.nC3+...+n2.nCn is

A
n(n+1).2n2
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B
n(n+1).2n1
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C
n(n1).2n2
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D
n(n1).2n1
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Solution

The correct option is A n(n+1).2n2
12.nC1+22.nC2+32.nC3+...+n2.nCn=nr=1(r2)nCrNow nr=1(r2)nCr=nr=1{r(r1)+r}nCr=nr=1r(r1).nCr+nr=1r.nCr=nr=2n(n1).n2Cr2+nr=1n.n1Cr1 (r/.n(n1)!r/(nr)!(r1)!)=n(n1)nr=2n2Cr2+nnr=1n1Cr1=n(n1).2n2+n.2n1=n.2n2(n1+2)=n(n+1).2n2

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