[POTD] Lucky Kapoor's Lucky 13

[MathBuster Forum]

Problem

Lucky Kapoor considers the number 13 and its multiples to be his lucky numbers. He creates a new board game in which a player needs to roll three non-standard fair dice: one with 8 faces, another with 12 faces, and a third with 20 faces. Each face has a different number on it, from 1 up to the number of faces on that die.

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What is the probability that the sum of the numbers on the top faces is divisible by 13?

Note: Please share your solution, not just the answer.