Lattice paths in 3D to the point (3,4,5)

Starting at the origin, you move through the integer lattice in three dimensions, and each move adds $1$ to exactly one of the $x$, $y$ or $z$ coordinates. How many different routes reach $(3,4,5)$?

$27{,}720$. Every route is 12 moves: 3 in $x$, 4 in $y$ and 5 in $z$, in some order. The number of such orders is $\tfrac{12!}{3!\,4!\,5!}=27{,}720$.