Mathematics - Binomial Theorem Question with Solution | TestHub

$\begin{array}{rl}& \sum _{\mathrm{r}=1}^{15}{\mathrm{r}}^2\left({}^{15}{\mathrm{C}}_{\mathrm{r}}\right)\Rightarrow 15\sum _{\mathrm{r}=1}^{15}{\mathrm{r}}^{14}{\mathrm{C}}_{\mathrm{r}-1}\\ & 15\sum _{\mathrm{r}=1}^{15}(\mathrm{r}-1+1){}^{14}{\mathrm{C}}_{\mathrm{r}-1}\\ & 15\cdot 14\sum _{\mathrm{r}=1}^{15}{}^{13}{\mathrm{C}}_{\mathrm{r}-2}+15\sum _{\mathrm{r}=1}^{15}{}^{14}{\mathrm{C}}_{\mathrm{r}-1}\\ & 15\cdot 14\cdot 2^{13}+15\cdot 2^{14}\\ & 3^1\cdot 2^{13}(70+10)\\ & 3^1\cdot 2^{13}\cdot 80\\ & 3^1\cdot 5^1\cdot 2^{17}\\ & \mathrm{m}=17\mathrm{n}=1\mathrm{k}=1\\ & \text{ option }(1)\end{array}$