The coefficient of  $x^{2012}$ in the expansion of ${(1-x)}^{2008}{(1+x+x^2)}^{2007}$  is equal to k then $\frac{k}{2}=$  ______

 $(1-x){(1-x)}^{2007}{(1+x+x^2)}^{2007} (1-x){(1-x^3)}^{2007} (1-x){(}^{2007}{C}_0{-}^{2007}{C}_1\left(x^3\right)+\mathrm{......})$
 General term
 $$\begin{gathered} (1-x)\left({(-1)}^{r2007}{C}_r{x}^{3r}\right) (-1){}^{r2007}C_r{x}^{3r}-(-1){}^{r2007}C_r{x}^{3r+1} \\ 3r=2012 r\ne \frac{2012}{3} 3r+1=2012 \\ 3r=2011 r\ne \frac{2011}{3} \end{gathered}$$
 Hence there is no term containing $x^{2012}$ .
So coefficient of  $x^{2012}=0$