${}^{15}\mathrm{C}_2+{2.}^{15}{\mathrm{C}}_3+3{.}^{15}{\mathrm{C}}_4+\dots .+14{.}^{15}{\mathrm{C}}_{15}=$

Sum $=\sum _{\mathrm{r}=2}^{15} (\mathrm{r}-1) {}^{15}{\mathrm{C}}_{\mathrm{r}}$

         $\begin{array}{rcl} & =& \sum _{\mathrm{r}=2}^{15} \mathrm{r} {}^{15}{\mathrm{C}}_{\mathrm{r}}-\sum _{\mathrm{r}=2}^{15} {}^{15}{\mathrm{C}}_{\mathrm{r}}\\ & =& \sum _{\mathrm{r}=2}^{15} \mathrm{r} \frac{15!}{(15-\mathrm{r}) ! \mathrm{r}!}-\left[2^{15}-\left({}^{15}{\mathrm{C}}_0 {}^{15}{\mathrm{C}}_1\right)\right]\\ & =& \sum _{\mathrm{r}=2}^{15} \frac{15!}{(15-\mathrm{r}) ! (\mathrm{r}-1) !}-2^{15}+16\\ & =& 15 \sum _{\mathrm{r}=2}^{15} {}^{14}{\mathrm{C}}_{\mathrm{r}-1}-2^{15}+16\\ & =& 15\left[2^{14}-1\right]-2^{15}+16\\ & =& \left(13\right) 2^{14}+1\end{array}$