In the expansion of ${(1 + x + x^{2})}^{- 3}$ , the coefficient of $x^{6}$ is

- 3

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3

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9

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- 9

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Solution

The correct option is B $3$
${(1 + x + x^{2})}^{- 3}$
$= {(\frac{1 - x^{3}}{1 - x})}^{- 3}$
$= {(\frac{1 - x}{1 - x^{3}})}^{3}$
$= (1 - x)^{3} (1 - x^{3})^{- 3}$
$= (1 - 3 x + 3 x^{2} - x^{3}) (1 + 3 x^{3} + \frac{(- 3) (- 4)}{2} (x^{3})^{2} + . . . .)$
$= (1 - 3 x + 3 x^{2} - x^{3}) (1 + 3 x^{3} + 6 x^{6} + . . . .)$
The coefficient of $x^{6}$ in the above expansion
$= (- 1) (3) + (1) 6$
$= 3$

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