If ${}^{20}C_r$ is the co-efficient of $X^r$ in the expansion of ${(1+x)}^{20},$ then the value of $\sum _{r=0}^{20}{r}^2 {}^{20}C_r is equal to :$

$\begin{array}{rcl}\sum _{\mathrm{r}=0}^{20} {\mathrm{r}}^2 {}^{20}{\mathrm{C}}_{\mathrm{r}}& =& \left[\left(\mathrm{r}(\mathrm{r}-1)\right)+\mathrm{r}\right] {}^{20}{\mathrm{C}}_{\mathrm{r}}\\ & =& \sum _{\mathrm{r}=0}^{20} \mathrm{r}(\mathrm{r}-1) {}^{20}{\mathrm{C}}_{\mathrm{r}}+\sum _{\mathrm{r}=0}^{20} \mathrm{r} {}^{20}{\mathrm{C}}_{\mathrm{r}}\\ & =& \left(20\right) (20-1) 2^{18}+20 2^{19}\\ & =& 2^{18} \left[400-20+40\right]\\ & & 420\times 2^{18}\end{array}$     $\boxed{\sum _{\mathrm{r}=0}^{20} \mathrm{r}(\mathrm{r}+1) {\mathrm{C}}_{\mathrm{r}}=\mathrm{n}(\mathrm{n}-1) 2^{\mathrm{n}-2}}$