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

Question

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

Answer: 0

Accepted Answer

$\begin{gathered} (1-x)(1-x{)}^{2007}{(1+x+{x}^{2})}^{2007} \ (1-x){(1-{x}^{3})}^{2007} \ (1-x)({}^{2007}{C}_{0}-{}^{2007}{C}_{1}({x}^{3})+\ldots \ldots .) \end{gathered}$

General term

$\begin{gathered} (1-x)((-1{)}^{r}{}^{2007}{C}_{r}{x}^{3r}) \ (-1{)}^{r2007}{C}_{r}{x}^{3r}-(-1{)}^{r2007}{C}_{r}{x}^{3r+1} \ 3r=2012 \ r
e \frac{2012}{3} \ 3r+1=2012 \ 3r=2011 \ r
e \frac{2011}{3} \end{gathered}$

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

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

Solutions · 1