2024 SSMO Accuracy Round Problems/Problem 6
Problem
Three six-sided dice are rolled. Then, the product of the three numbers on the top faces in calculated. If the probability of getting the product that is not a perfect square can be expressed as 
where 
and 
are relatively prime positive integers, what is 
?
Solution
For simplicity, we will count the number of ways of getting a product that is a perfect square. We will do casework on the number of perfect squares rolled. Firstly, there are 
ways to roll 3 perfect squares. Next, if two perfects squares rolled and the other isn't a perfect square, it is impossible for the final product to be a perfect square. If 1 perfect square is rolled, then the other two numbers rolled must be equal. There are 
ways to permute the rolls, 
perfect squares, and 
non-perfect squares, giving 
possibilities. Finally, if no perfect squares are rolled, we must have 
giving another 6 possibilities. In conclusion, the answer is ![\[1-\frac{8+24+6}{6^3} = 1-\frac{19}{108} = \frac{89}{108}\implies 89+108 = \boxed{197}.\]](http://latex.artofproblemsolving.com/2/c/5/2c5e844d72756ef755285e1d60ac8e89d1e260ae.png)
~SMO_Team
