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Question

Statement-1: nr=0(r+1)nCr=(n+2)2n1.
Statement-2: nr=0(r+1)nCrxr=(1+x)n+nx(1+x)n1.

A
Statemet -1 is false, Statement - 2 is ture
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B
Statemet -1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
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C
Statement -1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1
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D
Statement -1 is true, Statement -2 is false
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Solution

The correct option is B Statemet -1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
We have
nr=0(r+1)nCrxr=nr=0rnCrxr+nr=0nCrxr=nr=1 r.nrn1Cr1xr+(1+x)n=nxnr=1n1Cr1xr1+(1+x)n=nx(1+x)n1+(1+x)n=RHS
Statement 2 is correct.
Putting x=1, we get
nr=0(r+1)ncr=n.2n1+2n=(n+2).2n1
Statement 1 is also true and statement 2 is a correct explanation for statement 1.

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