Help using binomial theorem to prove the following

Question

Use the binomial theorem to prove that $C(n,0) + C(n,1)\cdot 2 + C(n,2)\cdot 2^2 + \dots + C(n,n)\cdot 2^n = 3^n.$

I'm pretty confused with this concept, so any help would be greatly appreciated.

Answer 1

Binomial Theorem says that for $n\in\mathbb{N}$ $$(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\cdots+\binom{n}{n-1}x^{n-1}+\binom{n}{n}x^{n}$$ for all $x\in\mathbb{R}$.

Put $x=2$.