Reading through my textbook I came across the following problem, and I am looking for some help solving it. In the back of the textbook they have shown the final answer but I'm not quite sure how to get there. If I could get some help that would be great, thanks!
Find, $$\sum_{k=0}^{n}4^k \binom{n}{k}$$
The solution given is,
$$C(7,1)C(4,1)+C(7,2)C(4,2)+C(7,3)C(4,3)+C(7,4)C(4,4)$$
Consider the expansion $(1+x)^n$ using binomial theorem. $$(1+x)^n = \sum_{k=0}^{n} x^k \binom{n}{k}$$ Then substitute $x = 4$. That should lead you to the answer $5^n$.