Question

$\left(\begin{array}{c}n\\ 0\end{array}\right)+2\left(\begin{array}{c}n\\ 1\end{array}\right)+2^2\left(\begin{array}{c}n\\ 2\end{array}\right)+...+2^n\left(\begin{array}{c}n\\ n\end{array}\right)$ is equal

• A. $2^n$
• B. $0$
• C. $3^n$
• D. $noneofthese$

Answer 1

The correct option is A $2^n$

${}^n{C}_0+2^n{C}_1+2^2{}^n{C}_n+......+2^n{}^n{C}_n$

$(1+x{)}^n{=}^n{C}_0{+}^n{C}_{1}x{+}^n{C}_2{x}^2{+}^n{C}_3{x}^3....{.}^{n}C-n{x}^n$

$x=1\Rightarrow (1+1{)}^n$

$\Rightarrow 2^n$