Question

If $C_{0}, C_{1}, C_{2}, \ldots, C_{15}$ are the binomial coefficients in the expansion of $(1+x)^{15}$, then $\frac{C_{1}}{C_{0}}+\frac{2 C_{2}}{C_{1}}+\frac{3 C_{3}}{C_{2}}+\ldots+\frac{15 C_{15}}{C_{14}}$ is equal to

• A. 32
• B. 64
• C. 128
• D. None of these

Answer 1

Answer: (D)

$ \frac{C_{1}}{C_{0}}+\frac{2 C_{2}}{C_{1}}+\frac{3 C_{3}}{C_{2}}+\ldots+\frac{15 C_{15}}{C_{14}} $
$=n+\frac{\frac{2 n(n-1)}{2}}{n}+\frac{\frac{3 n(n-1)(n-2)}{3 \times 2}}{\frac{n(n-1)}{2}}+\ldots $ upto $15$ terms
$ = n+(n-1)+(n-2)+\ldots+$ upto $ 15 $ terms
$= \frac{15(15+1)}{2}=120 $