Transcendental Dice Rolls

Probability Level 5

Three fair 6-sided dice each have their sides labeled 0,1,e,π,i,20,1,e,\pi, i, \sqrt{2}. If these dice are rolled, the probability that the product of all the numbers on the top face is real can be expressed as ab \frac {a}{b}, where aa and bb are coprime positive integers. What is the value of a+ba+b?

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