The value of $\sum _{\mathrm{r}=1}^{10} {\mathrm{r}}^2\cdot \frac{{ }^{26}{\mathrm{C}}_{\mathrm{r}}}{{ }^{26}{\mathrm{C}}_{\mathrm{r}-1}}$ is

$\begin{array}{c}\mathrm{S}=\sum _{\mathrm{r}=1}^{10} {\mathrm{r}}^2\cdot \frac{{ }^{26}{\mathrm{C}}_{\mathrm{r}}}{{ }^{26}{\mathrm{C}}_{\mathrm{r}-1}}=\sum _{\mathrm{r}=1}^{10} {\mathrm{r}}^2\cdot \left(\frac{27-\mathrm{r}}{\mathrm{r}}\right)=\sum _{\mathrm{r}=1}^{10} \left(27\mathrm{r}-{\mathrm{r}}^2\right)\\ =\frac{27\mathrm{r}(\mathrm{r}+1)}{2}-\frac{\mathrm{r}(\mathrm{r}+1)\cdot (2\mathrm{r}+1)}{6}\end{array}$

Putting r = 10

$\therefore \mathrm{S}=27\left(5\right)\left(11\right)-\left(5\right)\left(11\right)\left(7\right)=1100$