Show that the sum $\sum_{k = 0}^n 2^k \binom{n}{k}$ is equal to $3^n$

Question

How can I show that the sum

$$ \sum_{k = 0}^n 2^k \binom{n}{k}$$

is equal to $3^n$?

Answer 1

For a combinatorial interpretation: suppose you have $n$ distinguishable eggs, and you want to paint some of them red, some of them blue, and leave some of them unpainted. Show that both sides of your desired equation count the number of ways to do this.