Question

If $C_0,C_1,C_2,.....,C_{20}$ are the binomial coefficients in the expansion of ${\left(1+x\right)}^{20},$ then the value of $\frac{C_1}{C_0}+2\frac{C_2}{C_1}+3\frac{C_3}{C_2}+.....+19\frac{C_{19}}{C_{18}}+20\frac{C_{20}}{C_{19}}$ is equal to (where $C_r$ represents $\_{}^n{C}_r^{}$ )

• A. $120$
• B. $210$
• C. $180$
• D. $240$

Answer 1

Answer: (B)

$\frac{C_r}{C_{r-1}}=\frac{n-r+1}{r}=\frac{20-r+1}{r}=\frac{21-r}{r}$
$\Rightarrow r\cdot \frac{C_r}{C_{r-1}}=\left(21-r\right)$
$\Rightarrow \sum _{r=1}^{20}r\cdot \frac{C_r}{C_{r-1}}=\sum _{r=1}^{20}\left(21-r\right)$
$=21\sum _{r=1}^{20}1-\sum _{r=1}^{20}r$
$=21\times 20-\frac{20\times 21}{2}$
$=210$