JEE Main2008MathematicsBinomial TheoremActual
Statement-1: _ r=0 ^n(r+1)^n C_r=(n+2) 2^ n-1 Statement -2: _ r=0 ^n(r+1)^n C_r x^r=(1+x)^n+n x(1+x)^ n-1 .
Options
- AStatement -1 is false, Statement -2 is true
- BStatement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement -1
- CStatement -1 is true, Statement -2 is true; Statement -2 is not a correct explanation for Statement -1 .
- DStatement -1 is true, Statement -2 is false.
Correct answer
B. Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement -1
Step-by-step solution
aligned & _ r=0 ^n(r+1) ^n C_r= _ r=0 ^n r ^n C_r+ ^n C_r & = _ r=0 ^n r n r n-1 r-1 + _ r=0 ^n ^n C_r=n 2^ n-1 +2^n & =2^ n-1 (n+2) aligned Statement -1 is true aligned & (r+1)^n C_r x^r= r ^n C_r x^r+ ^n C_r x^r & =n _ r=0 ^n n-1 r-1 x^r+ _ r=0 ^n ^n C_r x^r=n x(1+x)^ n-1 +(1+x)^n aligned Substituting x=1 (r+1)^n C_r=n 2^ n-1 +2^n Hence Statement -2 is also true and is a correct explanation of Statement -1 .