Question

The coefficient of $x^{10}$ in the expansion of $1+\left(1+x\right)+\ldots+\left(1+x\right)^{20}$ is

• A. $^{19}C_{9}$
• B. $^{20}C_{10}$
• C. $^{21}C_{11}$
• D. $^{22}C_{12}$

Answer 1

Answer: (C)

$^{21}C_{11}$

Answer 2

The correct option is(C): $^{21}C_{11}$

The given series is in GP. Hence, its sum
S=\frac{1\left\\{(1+x)^{20+1}-1\right\\}}{(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 $(1+x)^{21}-1$
$={ }^{21} C_{11}$