Question

If x + y = 1, then ${\sum }_{r=0}^{n}r{}^n{C}_r{x}^r{y}^{n-r}$ equals:

• A. nx
• B. ny
• C. n
• D. None

Answer 1

The correct option is A

nx

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

= ${}^n{C}_{1}x{y}^{n-1}+2.{}^n{C}_2{x}^2{y}^{n-2}+3.{}^n{C}_3{x}^3{y}^{n-3}$

+........+n .${}^n{C}_n{x}^n$

= nx $[y^{n-1}+(n-1)x{y}^{n-1}+\frac{(n-1)(n-2)}{2}{x}^2{y}^{n-3}+............+x^{n-1}]$

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