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Question

The value of C01.3 - C12.3+ C23.3- C34.3+ ..................+(1)n Cn(n+1)3

A
3n+1
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B
n+33
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C
13(n+1)
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D
None of these
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Solution

The correct option is C 13(n+1)
C01.3C12.3+C23.3C34.3+....+(1)nCn(n+1).3
LHS =1n2+n(n1)3×2!.......+(1)n1n+1
=13(n+1)[(n+1)(n1)n2(n1)(n+1)n3......(1)n1]
=13(n+1)[n+1C1n+1C2+n+1C3n+1C4+......(1)n.n+1Cn+1]
=13(n+1)[1(n+1C0n+1C1+n+1C2......(1)n.n+1Cn+1)]
=13(n+1)[1(11)n+1]
=13(n+1)[10]
=13(n+1)
Hence, the answer is 13(n+1).


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