The coefficient of $x^{2012}$ in the expansion of $(1-x{)}^{2008}{\left(1+x+x^2\right)}^{2007}$, is

We have

$\begin{array}{l}(1-x{)}^{2008}{\left(1+x+x^2\right)}^{2007}=(1-x){\left(1-x^3\right)}^{2007}\\ -{\left(1-x^3\right)}^{2007}-x{\left(1-x^3\right)}^{2007}\end{array}$

We find that $x^{2012}$ is of the form $x^{3r+2}$ where as ${\left(1-x^3\right)}^{2007}$

contains terms of the form $x^{3r}$ and $x{\left(1-x^3\right)}^{2007}$ contains terms of the form $x^{3r+1}$ So, there is no term containing $x^{2012}$ in the given expansion 

Hence, coefficient of $x^{2012}$ is zero.