The number of bit strings with only two occurrences of 01 How many bit strings of length $n$, where $n \geq 4$, contain exactly two occurrences of $01$?

Any such sequence can be written as:
$$1^k0^l1^m0^p1^q0^r$$
where $k+l+m+p+q+r=n$ and $l,m,p,q\geq 1$ and $k,r\geq 0$.
But we can see that as the number of ways of writing $n-4$ as the sum of $6$ non-negative integers, which is ${n+1}\choose{5}$.
