Sum of binomial coefficient equation

Question

I am trying to show the following equation

using binomial expansion but it's not getting me anywhere. How should I set it up?

I've tried expanding $(2+2)^{n}$ and using the fact that

but it's not quite right. Any hints?

Answer 1

Noting that the $4^n$ can be factored out of the left sum, we have that

$$\begin{align} \sum_{k=0}^n \binom{n}{k} \, 4^n &= 4^n \, \sum_{k=0}^n \binom{n}{k} \\[2mm] &= 4^n \cdot 2^n \\[2mm] &= 8^n \\[2mm] &= (1+7)^n \\[2mm] &= \sum_{k=0}^n \binom{n}{k} \, 7^k \end{align}$$