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  4. The coefficient of x10 in the expansion of 1+(1+x)+ ldots+(1+x)20 is

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

WBJEEWBJEE 2012Binomial Theorem

Solution:

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 $(1+x)^{21}-1$
$={ }^{21} C_{11}$