How many unit paths are there in R3

Question

How many unit paths are there in $\Bbb R^3$ from $(0,0,0)$ to $(n,n,n)$ that never pass below the plane $y=x$? This means that for every point $(a,b,c)$ on the path we have $a\le b$.

Answer 1

Hint: Do you know how to do it in $\Bbb R^2$? See the Catalan numbers. Then pick $n$ places of the path of $3n$ to put steps in the third axis. Why does this work?