If in the expansion of ${\left({\mathrm{x}}^3-1/{\mathrm{x}}^2\right)}^{\mathrm{n}}, \mathrm{n}\in \mathrm{N}$, sum of the coefficients of ${\mathrm{x}}^5\text{ and }{\mathrm{x}}^{10}\text{ is }0\text{, }$value of n is 5m. then the value of m, is

$$\begin{gathered} {\left({\mathrm{x}}^3-\frac{1}{{\mathrm{x}}^2}\right)}^{\mathrm{n}}\Rightarrow \mathrm{r}=\frac{\mathrm{np}-\mathrm{k}}{\mathrm{p}+\mathrm{q}}\Rightarrow \mathrm{k}=5\Rightarrow \mathrm{r}=\frac{3\mathrm{n}-5}{5} \\ \Rightarrow \mathrm{r}=\frac{3\mathrm{n}-5}{5}{;}^{\mathrm{n}}{{\mathrm{C}}_{\left(\frac{3\mathrm{n}-5}{5}\right)}}^{\mathrm{n}}{\mathrm{C}}_{\left(\frac{3\mathrm{n}-10}{5}\right)} \\ \Rightarrow \frac{3\mathrm{n}-5}{5}+\frac{3\mathrm{n}-10}{5}=\mathrm{n}\Rightarrow 6\mathrm{n}-15=5\mathrm{n} \\ \Rightarrow \mathrm{n}=15\Rightarrow 5\mathrm{m}=15\Rightarrow \mathrm{m}=3 \\ \begin{array}{l}\text{ D) }{2}^{2009}=2\left[2^{2008}\right]=2{\left[2^4\right]}^{502}=2[17-1{]}^{502}\\ =17\mathrm{k}+2(-1{)}^{502}=17\mathrm{K}+2\end{array} \end{gathered}$$