The coefficient of x^{7} in the expression (1 | vedclass

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(B) The given expression is a geometric series with first term $a = (1+x)^{10}$,common ratio $r = \frac{x}{1+x}$,and $n = 11$ terms.Sum $S = a \frac{1

Vedclass · Accepted Answer

$330$

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(B) The given expression is a geometric series with first term $a = (1+x)^{10}$,common ratio $r = \frac{x}{1+x}$,and $n = 11$ terms.Sum $S = a \frac{1-r^{n}}{1-r} = (1+x)^{10} \frac{1-(\frac{x}{1+x})^{11}}{1-\frac{x}{1+x}} = (1+x)^{10} \frac{1-\frac{x^{11}}{(1+x)^{11}}}{\frac{1+x-x}{1+x}} = (1+x)^{11} \left(1-\frac{x^{11}}{(1+x)^{11}}ight) = (1+x)^{11}-x^{11}$.We need the coefficient of $x^{7}$ in $(1+x)^{11}-x^{11}$.The general term in the expansion of $(1+x)^{11}$ is given by $^{11}C_{r} x^{r}$.For $r = 7$,the coefficient is $^{11}C_{7} = \frac{11!}{7!4!} = \frac{11 	imes 10 	imes 9 	imes 8}{4 	imes 3 	imes 2 	imes 1} = 330$.

The coefficient of $x^{7}$ in the expression $(1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}$ is

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