Coefficient of xr in the expansion of 1-x21-2x3 is

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Question

Coefficient of $x^{r}$ in the expansion of $\frac{(1 + x)^{2}}{(1 - 2 x)^{3}}$ is

(2 r^{2} + r + 2) 2^{r - 2}

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(9 r^{2} + 15 r + 8) 2^{r - 3}

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(9 r^{2} + 15 r + 8) 2^{r - 2}

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(2 r^{2} + r + 2) 2^{r - 3}

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Solution

The correct option is B $(9 r^{2} + 15 r + 8) 2^{r - 3}$
${(1 + x)}^{2}$ . ${(1 - 2 x)}^{- 2}$
$(1 + x^{2} + 2 x)$ .(.........+ $^{3 + r - 1} C_{r}$ ${(2 x)}^{r}$ + $^{3 + r - 2} C_{r - 1}$ . ${(2 x)}^{r - 1}$ + $^{3 + r - 3} C_{r - 1}$ . $(2 x)^{r - 2}$ ...........)
So, we have to find coefficent of $x^{r}$ that will be products as
$^{2 + r} C_{r}$ +2 $^{r + 1} C_{r - 1}$ + $^{r} C_{r - 2}$
$=$ $\frac{(r + 2)!}{r! .2!}$ $2^{r}$ + $2. \frac{(r + 1)!}{2! . (r - 1)!}$ $2^{r - 1}$ + $\frac{r!}{=} 2! (r - 2)!$ . $2^{r - 2}$
$= (r + 2) (r + 1)$ $2^{r - 1}$ $+ (r + 1) (r)$ $2^{r - 1}$ $+ (r) (r - 1)$ $2^{r - 3}$
$= (4 ($ $r^{2}$ $+ 3 r + 2 +$ $r^{2}$ $+ r) +$ $r^{2}$ $- r)$
$=$ $2^{r - 3}$ $(8$ $r^{2}$ $+ 16 r + 8 +$ $r^{2}$ $- r)$
$=$ $2^{r - 3}$ $(9$ $r^{2}$ $+ 15 r + 8)$

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Coefficient of xr in the expansion of (1+x)2(1−2x)3 is

Coefficient of $x^{r}$ in the expansion of $\frac{(1 + x)^{2}}{(1 - 2 x)^{3}}$ is