$x+y=1\Rightarrow {\displaystyle \sum _{r=0}^{n}r.{}^n{\text{C}}_r{x}^r{y}^{n-r}=}$

$={\displaystyle \sum _{r=0}^{n}r.\frac{n!}{\left(n-r\right)!r!}{x}^r{y}^{n-r}}$

$={\displaystyle \sum _{r=0}^{n}r.\frac{n\left(n-1\right)!}{\left(n-r\right)!r\left(r-1\right)!}{x}^r{y}^{n-r}}$

$=n.{\displaystyle \sum _{r=0}^n\frac{\left(n-1\right)!}{\left(n-1\right)-\left(r-1\right)!\left(r-1\right)!}{x}^r{y}^{n-r}}$

$=n.{\displaystyle \sum _{r=0}^n{}^{\left(n-1\right)}{C}_{r-1}.x^r.y^{n-r}}$

$=n.{\displaystyle \sum _{r=0}^n{}^{\left(n-1\right)}{C}_{r-1}.x^{r-1+1}.y^{\left(n-1\right)-\left(r-1\right)}}$

$=n.{\displaystyle \sum _{r=0}^n{}^{\left(n-1\right)}{C}_{r-1}.x^{r-1}.x{y}^{\left(n-1\right)-\left(r-1\right)}}$

$=nx{\displaystyle \sum _{r=0}^n{}^{\left(n-1\right)}{C}_{r-1}.x^{r-1}{y}^{\left(n-1\right)-\left(r-1\right)}}$

$=nx{\displaystyle \sum _{r=0}^n{\left(x+y\right)}^{r-1}}$

$=nx$