JEE MAIN - Mathematics (2024 - 27th January Evening Shift - No. 22)

The coefficient of $$x^{2012}$$ in the expansion of $$(1-x)^{2008}\left(1+x+x^2\right)^{2007}$$ is equal to _________.
Answer
0

Explanation

$$\begin{aligned} & (1-x)(1-x)^{2007}\left(1+x+x^2\right)^{2007} \\ & (1-x)\left(1-x^3\right)^{2007} \\ & (1-x)\left({ }^{2007} C_0-{ }^{2007} C_1\left(x^3\right)+\ldots \ldots .\right) \end{aligned}$$

General term

$$\begin{aligned} & (1-x)\left((-1)^r{ }^{2007} C_r x^{3 r}\right) \\ & (-1)^{r 2007} C_r x^{3 r}-(-1)^{r 2007} C_r x^{3 r+1} \\ & 3 r=2012 \\ & r \neq \frac{2012}{3} \\ & 3 r+1=2012 \\ & 3 r=2011 \\ & r \neq \frac{2011}{3} \end{aligned}$$

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

So coefficient of $$\mathrm{x}^{2012}=0$$

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