$\text{ The value of }\frac{\sum _{r=2}^{10}{ }^r{C}_2{\cdot }^{10}{C}_r}{10}\text{ is }$

$\begin{array}{l}\frac{1}{10}\left(\sum _{\mathrm{r}=2}^{10} \frac{\mathrm{r}(\mathrm{r}-1)}{2}\cdot \frac{10!}{\mathrm{r}!(10-\mathrm{r})!}\right)\\ =\frac{1}{10}\left(\sum _{\mathrm{r}=2}^{10} \frac{1}{2}\cdot \frac{10!}{(\mathrm{r}-2)!(10-\mathrm{r})!}\right)\\ =\frac{1}{10}\left(45\sum _{\mathrm{r}=2}^{10} \frac{8!}{(\mathrm{r}-2)!(10-\mathrm{r})!}\right)\\ =\frac{1}{2}\left(9\sum _{\mathrm{r}=2}^{10}{ }^8{\mathrm{C}}_{\mathrm{r}-2}\right)\\ =\frac{1}{2}\left(9\times \left({ }^8{\mathrm{C}}_0{+}^8{\mathrm{C}}_1+\dots ..{+}^8{\mathrm{C}}_8\right)\right)\\ =\frac{1}{2}\times 9\times 2^8\\ =1152\end{array}$