Question

If $x+y=1$, Find ${\sum }_{r=1}^{n}r\stackrel{n}{\underset{r}{C}}{x}^r{y}^{n-r}$

Answer 1

We have,

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

$\Rightarrow nx{\sum }_{r=1}^n{}^{n-1}{C}_{r-1}{x}^{r-1}{y}^{(n-1)-(r-1)}$

$\Rightarrow nx(x+y{)}^{n-1}$

$\Rightarrow nx$ [since $x+y=1$]

$\Rightarrow {\sum }_{r=0}^{n}r{}^n{C}_r{x}^r{y}^{n-r}=nx$