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Hi Bruno, I saw your probability challenge and decided to solve it by brute force. Out of all 7776 possible rolls of five six-sided dice, 1656 outcomes (about 21.30%) will have three or more dice with the same face. I'll post my work & equation soon.
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Here's the general equation to find the probability that at least x out of y n-sided dice have the same face. https://www.wolframalpha.com/input?i=calculate+%5Csum_%7Bi+%3D+x%7D%5Ey+%28n+%5Cbinom%7By%7D%7Bi%7D+%281%2Fn%29%5Ei+%28%28n-1%29%2Fn%29%5E%7By-i%7D%29+when+x+%3D+3%2C+y%3D+5%2C+n+%3D+6
HOLY SHIT I never thought anyone would care to do that thank you a lot! I'll definitely use your equasion while still testing my card game, you're awesome!