I have to prove the identity using a combinatorial proof:
$\displaystyle\sum\limits_{k=0}^n 2^k \binom{n}{k} = 3^n$
I think this should be my combinatorial proof:
We want to form a committee of $k$ people from a total of $n$ people. There are two ways of counting this committee.
1) Go through each member from the $n$ total people, and decide if they should be added to the committee or not, until we have reached $k$ people. This gives us the LHS.
...For the RHS, however, I am not sure how to form it. I think it should be something like forming subsets of $3$ people and choosing from that, but I'm not sure how that will form the same committee as the LHS.
EDIT: Okay, I also had the idea of forming a ternary string, and I could get the RHS this way. But I was not sure about the LHS. But the first answer gave me the right idea. Thanks a lot.