Six standard 6-sided dice are rolled, and the resulting numbers are multiplied together. What is the probability that the product is divisible by 125?

1453/23328

Step-by-step explanation:

Since we are going to have at least three 5's to obtain a number divisible by 125, the number of ways we can combine exactly three 5's is ⁶C₃. The number of ways we can combine the remaining dice without containing a five is 5³, since there are 3 remaining dice. So, for exactly three 5's , we have ⁶C₃ × 5³. Similarly, for exactly four 5's, we have ⁶C₄ × 5² (5² since there are two dice left). Also, for exactly five 5's, we have ⁶C₅ × 5¹ (5¹ since there are one die left). Finally, for exactly six 5's, we have ⁶C₆ × 5⁰(5⁰ since there are no dice left without a five).

So, for at least three 5's, we have ⁶C₃ × 5³ + ⁶C₄ × 5² + ⁶C₅ × 5¹ + ⁶C₆ × 5⁰

= 20 × 125 + 15 × 25 + 6 × 5 + 1 × 1

= 2500 + 375 + 30 + 1

= 2906 ways

The total number of ways of combining the six dice is 6⁶

So, the probability of obtaining a product divisible by 6 is thus P = 2906/6⁶ = 2906/46656

= 1453/23328

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