Question

The coefficient of $x^{10}$ in the expansion of $1+(1+x)+...............................+(1+x{)}^{20}$ is

• A. ${}^{19}{C}_9$
• B. ${}^{20}{C}_{10}$
• C. ${}^{21}{C}_{11}$
• D. ${}^{22}{C}_{12}$

Answer 1

The correct option is C ${}^{21}{C}_{11}$

The given series is in GP. Hence, its sum
$S=\frac{1\left\{\right(1+x{)}^{20+1}-1\}}{(1+x)-1}=\frac{(1+x{)}^{21}-1}{x}$
Therefore, the required coefficient of $x^{10}$ in the expansion of $\frac{(1+x{)}^{21}-1}{x}$
$=$ Coefficient of $x^{11}$ in the expansion of $(1+x{)}^{21}-1$
$={}^{21}{C}_{11}$