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Combination

If | x | < ...

Question

If $| x | < 1,$ then the coefficient of $x^{6}$ in the expansion of $(1 + x + x^{2})^{- 3}$ is $-$

A

3

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B

6

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C

9

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D

12

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Solution

The correct option is C $3$
$(1 + x + x^{2})^{- 3}$
$= [\frac{1}{1 + x + x^{2}}]^{3}$
$= [\frac{1 - x}{1 - x^{3}}]^{3}$
$= (1 - x)^{3} (1 - x^{3})^{- 3}$
$= (1 - 3 x + 3 x^{2} - x^{3}) (1 + 3 x^{3} + 6 x^{6} + 10 x^{9} . . . \infty)$ .
Hence coefficient of $x^{6}$ will be
$- 3 + 6$
$= 3$

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