Maths Olympiad Prep

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Combinatorics Difficulty 4.4 AIME Find the answer

In one step, one can move from the point A(ijk)A(i|j| k), with i,j,k{0,1,2,3}i, j, k \in\{0,1,2,3\}, to another point of the cube grid by always increasing exactly one of the coordinates by 1.
Determine the number of all shortest paths that lead from the origin O(000)\mathrm{O}(0|0| 0) to the point P(333)\mathrm{P}(3|3| 3).

Solution

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.