If ${ }^{30}{\mathrm{C}}_{\mathrm{r}}$ is the coefficient of x^r in the expansion of $(1+\mathrm{x}{)}^{30}$ , then the value of $\sum _{\mathrm{r}=1}^{30} r^{2 }\left({ }^{30}{\mathrm{C}}_{\mathrm{r}}\right)$ is equal to:

$\begin{array}{l}\mathrm{S}=\sum _{\mathrm{r}=1}^{30} {\mathrm{r}}^2{\times }^{30}{\mathrm{C}}_{\mathrm{r}}=30\sum _{\mathrm{r}=1}^{30} \mathrm{r}\left({ }^{ 29}{\mathrm{C}}_{\mathrm{r}-1}\right)\\ \mathrm{S}=30\left[\sum _{\mathrm{r}=1}^{30} {(\mathrm{r}-1)}^{29}{\mathrm{C}}_{\mathrm{r}-1}+\sum _{\mathrm{r}=1}^{30}{ }^{29}{\mathrm{C}}_{\mathrm{r}-1}\right]\\ \mathrm{S}=30\left[\sum _{\mathrm{r}=1}^{30} 29{\times }^{28}{\mathrm{C}}_{\mathrm{r}-2}+2^{29}\right]=30\left[29\times 2^{28}+2^{29}\right]\\ S=2^{29}\left(30+15\left(29\right)\right)=2^{29}\left(465\right)\end{array}$