IN-TST 2010 Problem 6

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$ s and $1'$ s such that there are equal number of $00$ and $11$ blocks in each string is equal to $$2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}$$