3D Moves II

How many paths are there from $(0,0,0)$ to $(x,y,z)$ in $3-$space, where $x,y,$ and $z$ are all positive integers, such that each movement in the path is a movement forward, right, or upwards one unit, except one movement downwards ($-z$ direction)? Solve this when $(x,y,z) = (2,4,6)$.