The sum of the series $\frac{{ }^{101}{\mathrm{C}}_1}{{ }^{101}{\mathrm{C}}_0}+\frac{2{\cdot }^{101}{\mathrm{C}}_2}{{ }^{101}{\mathrm{C}}_1}+\frac{3{\cdot }^{101}{\mathrm{C}}_3}{{ }^{101}{\mathrm{C}}_2}+\dots +\frac{{101}^{.101}{\mathrm{C}}_{101}}{{ }^{101}{\mathrm{C}}_{100}}$ is __________.

Here, ${\mathrm{T}}_{\mathrm{r}}=\frac{\mathrm{r}{\cdot }^{101}{\mathrm{C}}_{\mathrm{r}}}{{ }^{101}{\mathrm{C}}_{\mathrm{r}-1}}=\frac{\mathrm{r}\cdot (101-\mathrm{r}+1)}{\mathrm{r}}=102-\mathrm{r}$

$\begin{array}{l}\therefore \text{ Given sum }=\sum _{\mathrm{r}=1}^{101} (102-\mathrm{r})\\ =101+100+99+\dots +2+1\\ =5151\end{array}$