discrete math binomial theorem problem: computing the sum

Question

Compute the following sum:

$$\binom{n}{0} + 2\binom{n}{1} + 4\binom{n}{2} + 8\binom{n}{3} + \cdots + 2^n\binom{n}{n}$$

I know this question involves the binomial theorem, but I am not sure what to do.

Answer 1

This is just the Binomial Theorem. $$\sum_{k=0}^{n}2^k{n \choose k} = \sum_{k=0}^{n}1^{n-k}2^k{n \choose k} = (1+2)^n = 3^n $$