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Question

If nC0,nC1,nC2,,nCn denote the binomial coefficients in the expansion of (1+x)n and p+q=1, then nr=0r2 nCr pr qnr is

A
np
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B
npq
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C
n2p2+npq
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D
None of these
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Solution

The correct option is C n2p2+npq
We have nr=0r2 nCr pr qnr
=nr=0[r(r1)+r] nCr pr qnr
=nr=0r(r1) nCr pr qnr+nr=0r nCr pr qnr
=nr=0r(r1)nrn1r1 n2Cr2 pr qnr+nr=0rnr n1Cr1 pr qnr
=n(n1)p2(q+p)n2+np(q+p)n1
=n(n1)p2+np [q+p=1]
=n2p2np2+np
=n2p2+npq [q+p=1]

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