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Question

If n is an integer greater than 1, then anC1(a1)+nC2(a2)....+(1)n(an)=

A
a
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B
0
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C
a2
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D
2n
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Solution

The correct option is B 0
anC1(a1)+nC2(a2)....+(1)n(an)
=nr=0(1)rnCr(ar)
=nr=0(1)rnCranr=0(1)rnCrr
=anr=0(1)rnCrnr=1(1)rnCrr (when r=0, we get 0)
=a(11)nnr=1(1)rnrn1Cr1r
(nCr=nrn1Cr1) ((1+x)n=nr=0nCrxr when x=1 (11)r=nr=0nCr(1)r)
=a(0)nr=1(1)rnn1Cr1
=0nnr=1(1)rn1Cr1
=n[n1C0+n1C1n1C2(1)n1n1Cn1]
=n[0]=0 [(11)n1=n1C0n1C1+n+1C2n1C3+.....=(1)nn1Cn1=0].

1201351_1196321_ans_c3522b1090b4466d9049c6e5c137d226.jpg

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