Question

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

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is C

$(1+x{)}^n{=}^n{C}_0+x{.}^n{C}_1+x^2{.}^n{C}_2+\cdots +x^n{.}^n{C}_n$
$\text{put x=2}$
$\Rightarrow 3^n{=}^n{C}_0+{2.}^n{C}_1+2^2{.}^n{C}_2+2^3{.}^n{C}_3+\cdots +2^n{.}^n{C}_n$