Coefficient of ${\mathrm{x}}^{2009}$ in 

 ${\left(1+\mathrm{x}+{\mathrm{x}}^2+{\mathrm{x}}^3+{\mathrm{x}}^4\right)}^{1001}{(1-\mathrm{x})}^{1002}$   is

${\left(1+\mathrm{x}+{\mathrm{x}}^2+{\mathrm{x}}^3+{\mathrm{x}}^4\right)}^{1001}{(1-\mathrm{x})}^{1002}$

$=\left(1-\mathrm{x}\right){\left(1-{\mathrm{x}}^5\right)}^{1001},$
 so all the powers of x will be of the 5m or 5m + 1 $\left(\mathrm{m}\in I\right)$

So coff. of ${\mathrm{x}}^{2009}$ will be 0