This is a question that was asked at the start of the section on Catalan Numbers in my book. I'm having trouble answering it.
My Work
All of the legal paths (ones which do not cross over $y=x$) must begin with a move right and end with a move up. Further, any time the number of upward moves exceeds the number of downward moves, the path becomes illegal. So it would seem, the answer is the total number of paths from $(0,0)$ to $(n,n)$ - (Total illegal paths). The number of paths from $(0,0)$ to $(n,n)$ is $\binom{2n}{n}$. The number of illegal paths I'm slightly less clear on. How do I compute the number of illegal paths?