Question

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

• A. $1$
• B. $n$
• C. $nx$
• D. $ny$

Answer 1

Answer: (C)

$nx$

Expanding, we get
${}^n{C}_{1}x{y}^{n-1}+2{}^n{C}_2{x}^2{y}^{n-2}+3{}^n{C}_3{x}^3{y}^{n-3}+...n{}^n{C}_n{x}^n$
$=x[{}^n{C}_1{y}^{n-1}+2{}^n{C}_2{x}^1{y}^{n-2}+3{}^n{C}_3{x}^2{y}^{n-3}+...n{}^n{C}_n{x}^{n-1}]$
$=x\left[\frac{d(y+x{)}^n}{dx}\right]$ ...(keeping $y$ constant.)
$=x\left(n\right(x+y{)}^{n-1})$
Now $x+y=1$,
Hence we get
$x\left(n\right)$
$=nx$