What is the error in the following solution to the question: In how many strings of length $n$ composed of $\{0,1\}$ there's at least $m$ zeros?
Solution: choose $m$ places for the zeros, in the rest of the places choose either $0$ or $1$ therefore: $\binom n m 2^{n-m}$.
From trying out small numbers I can see that it's wrong but I don't see why it's wrong to choose places for the zeros other than maybe making a distinction of 'first' zeros, dividing by $m!$ to cancel that isn't right either.