The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}\left(1+x+x^2\right)^{2007}$ is equal to _________.
Answer (integer)
0
Solution
<p>$$\begin{aligned}
& (1-x)(1-x)^{2007}\left(1+x+x^2\right)^{2007} \\
& (1-x)\left(1-x^3\right)^{2007} \\
& (1-x)\left({ }^{2007} C_0-{ }^{2007} C_1\left(x^3\right)+\ldots \ldots .\right)
\end{aligned}$$</p>
<p>General term</p>
<p>$$\begin{aligned}
& (1-x)\left((-1)^r{ }^{2007} C_r x^{3 r}\right) \\
& (-1)^{r 2007} C_r x^{3 r}-(-1)^{r 2007} C_r x^{3 r+1} \\
& 3 r=2012 \\
& r \neq \frac{2012}{3} \\
& 3 r+1=2012 \\
& 3 r=2011 \\
& r \neq \frac{2011}{3}
\end{aligned}$$</p>
<p>Hence there is no term containing $\mathrm{x}^{2012}$.</p>
<p>So coefficient of $\mathrm{x}^{2012}=0$</p>
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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