The coefficient of x $^{11}$ in the expansion of ( $1 - 2$ x + $3 x^{2}$ )( $1$ + x) $^{11}$ is

164

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144

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116

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none of these

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Solution

The correct option is B $144$
$\to (1 - 2 x + 3 x^{2}) (1 + x)^{11}$

$= (3 x^{2} - 2 x + 1) (1 + x)^{11}$

General term of $(1 + x)^{11}$
$T_{r + 1} =^{11} C_{r} . x^{r}$
with $3 x^{2}, r = 9 \Rightarrow c o e f f . = (^{11} C_{9} \times 3)$
with $- 2 x, r = 10 \Rightarrow c o e f f . = - 2 \times^{11} C_{10}$
with $1, r = 11 \Rightarrow c o e f f =^{11} C_{11}$

$\Rightarrow c o e f f . =^{11} C_{9} \times 3 - 2 \times 11 + 1$
$= 11.5.3 - 22 + 1$

$= 11 (5 - 2) + 1 = (11 \times 13) + 1$

$= 143 + 1 = 144 \to (B)$

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