The Algebra and Combinatorics of PIE

Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x,$ $y,$ or $z$-directions. However, the evil David anticipated Justin's arrival, and so he has surreptitiously placed a mine at the point $(2, 2, 2).$ If at any point Justin is at most one unit away from this mine (in any direction), the mine detects his presence and explodes, thwarting Justin. How many paths can Justin take to reach his destination safely? (NIMO)