Coefficient of $x^{10} in (1 + 2 x)^{21} + (1 + 2 x)^{22} + \dots \dots + (1 + 2 x)^{30}$ is

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$^{31} C_{11}$

$2^{10} (^{31} C_{11} -^{21} C_{11})$

$2^{9} (^{31} C_{11} -^{21} C_{11})$

$2^{10} (^{30} C_{11} -^{21} C_{11})$

answer is A.

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Detailed Solution

Coefficient of $x^{10}$ in

$\begin{array}{l} (1 + 2 x)^{21} + (1 + 2 x)^{22} + \dots \dots + (1 + 2 x)^{30} \\ \Rightarrow= \frac{(1 + 2 x)^{21} [(1 + 2 x)^{10} - 1]}{2 x} \\ Cos x^{10} in \frac{(1 + 2 x)^{31} - (1 + 2 x)^{21}}{2 x} \Rightarrow coefficient of x^{11} in \\ \frac{1}{2} [(1 + 2 x)^{31} - (1 + 2 x)^{21}] \\ \Rightarrow= \frac{1}{2} [^{31} C_{11} 2^{11} -^{21} C_{11} 2^{11}] = (^{31} C_{11} -^{21} C_{11}) 2^{10} \end{array}$