The coefficient of x 2012 in the expansion of ( 1 - x ) 2008 ( 1 + x + x 2 ) 2007 is equal to _______ . [2024]

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Solution

Q.
The coefficient of $x^{2012}$ in the expansion of ${(1 - x)}^{2008} {(1 + x + x^{2})}^{2007}$ is equal to _______ . [2024]

Ans.
(0)

We have, ${(1 - x)}^{2008} {(1 + x + x^{2})}^{2007}$

$= {(1 - x)}^{2007} (1 - x) {(1 + x + x^{2})}^{2007} = (1 - x) {(1 - x^{3})}^{2007}$

$\Rightarrow (1 - x) ({}^{2007}{C_{0} - {}^{2007}{C_{1}} (x^{3}) + . . .})$

General term is $(1 - x) ({(- 1)}^{r} {}^{2007}{C_{r}} x^{3 r})$

$= {(- 1)}^{r} {}^{2007}{C_{r}} x^{3 r} - {(- 1)}^{r} {}^{2007}{C_{r}} x^{3 r + 1}$

Now, $3 r = 2012 \Rightarrow r \neq \frac{2012}{3}$

So, $3 r + 1 = 2012 \Rightarrow 3 r = 2011 \Rightarrow r \neq \frac{2011}{3}$

Hence, there is no term containing $x^{2012} .$

So, coefficient of $x^{2012} = 0$

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