Find the coefficient of $x^{6}$ in $(1 + x + x^{2})^{-} 3$

2

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Solution

The correct option is C $3$
$(1 + x + x^{2})^{- 3} = {\frac{(1 - x) (1 + x + x^{2})}{1 - x}}^{- 3}$

$= {\frac{(1 - x^{3})}{1 - x}}^{- 3} = {(\frac{1 - x}{1 + x^{3}})}^{3}$
$= (1 - x)^{3} (1 - x^{3})^{- 3}$
$= [1 - 3 x + 3 x^{2} - x^{3}] [1 + 3 x^{3} + 6 x^{6} + . . . . .]$

$= 1 - 3 x + 3 x^{2} - x^{3} + 3 x^{3} - 9 x^{4} + 9 x^{5} - 3 x^{6} + 6 x^{6} - ----------- - + . . . . . .$

$∴$ Coefficient of $x^{6} = - 3 + 6 = 6 - 3 = 3$ .

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