Coefficient of  $x^{2009}$  in the expansion of  ${(1+x+x^2+x^3+x^4)}^{1001}{(1-x)}^{1002}$is

${(1+x+x^2+x^3+x^4)}^{1001}{(1-x)}^{1002}$

${\left(\frac{1-x^5}{1-x}\right)}^{1001}{\left(1-x\right)}^{1002}={\left(1-x^5\right)}^{1001}\left(1-x\right)$

in the above expansion all the terms are in the form of $x^{5m} or x^{5m+1}$

So coefficient of $x^{2009}$ is 0