Let $X_n$ be the set of all word of the length $2 n$ over the alphabet $\{A,B\}$ which contain as many A's as B's.
The amount of elements of $X_n$ is $\displaystyle \binom{2n}{n}$, but why?
I thought about it for a long time but really am a bit slow today. Does anybody have a (simple) combinatorical explanation why this applies?
Thanks in advance!