The coefficient of x^2012 in the expansion of (1-x)^2008(1+x+x^2)^2007 is equal to _________.

(1-x)(1-x)^2007(1+x+x^2)^2007 \ (1-x)(1-x^3)^2007 \ (1-x)( ^2007 C_0- ^2007 C_1(x^3)+ ... ... .) \ General term (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 \ 3 r=2012 \ r ≠ (2012 / 3) \ 3 r+1=2012 \ 3 r=2011 \ r ≠ (2011 / 3) \ Hence there is no term containing x^2012. So coefficient of x^2012=0

The coefficient of x2012 in the expansion of 1x20081xx2... | null (Mathematics)

\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 $\mathrm{x}^{2012}=0$

Explanation

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