Question

$(\frac{n}{0})+2(\frac{n}{1})+2^2(\frac{n}{2})+....+2^n(\frac{n}{n})$ is equal to

• A. $2^n$
• B. $0$
• C. $3^n$
• D. none of these

Answer 1

The correct option is C $3^n$

Given, $(\frac{n}{0})+2(\frac{n}{1})+2^2(\frac{n}{2})+....+2^n(\frac{n}{n})$

Simplifying the above expression we get
$={}^n{C}_0{2}^0+{}^n{C}_1{2}^1+{}^n{C}_2{2}^2+...{}^n{C}_n{2}^n$
Consider $x=2$ in the expansion of $(1+x{)}^n$.
$=(1+2{)}^n$
$=3^n$.