# How many binary strings of length $n$ contain $k$ flips?

If I have say the string $$1010010001010101$$, which has a length of $$16$$ and there are $$12$$ flips. My thoughts are to just count the number of ways I can stick a $$10$$ in the there so $${n-1 \choose 0.5k}$$ but I know that doesn't account for the times the first and last bits aren't the same and $$k$$ is odd, or what the bits in between the $$10$$ are, since they could be a run of $$1$$s or $$0$$s.

Been out of school for a couple of years and it's kind of depressing how much I've forgotten, so any help would be greatly appreciated, thanks

If you have $$n$$ digits, then there are $$n-1$$ spaces between the digits, of which $$k$$ are "flips" and the others are not. We have $$\binom{n-1}{k}$$ ways to position the flips. Additionally, for each of these choices of flip positions, the leftmost digit could be a zero or a one, so we multiply the number of choices by 2 to account for this difference: $$2 \binom{n-1}{k}$$