Three different dice are rolled three times. The Probability that they show different numbers only two times is:
Step-by-Step Solution
Key Concept: Recognize that 'different numbers only two times' means exactly 2 out of 3 rolls show all different faces. Use binomial selection: choose which 2 rolls have different faces, then multiply P(different)² × P(not all different) = C(3,2) × (5/9)² × (4/9).
Probability that 3 different dice show different faces is $\frac{P_6}{6^3} = \frac{6!/(6-3)!}{6^3} = \frac{120}{216} = \frac{5}{9}$. Using conditional probability with at least one pair matching: $\binom{3}{2}\left(\frac{5}{9}\right)^2 \times \left(\frac{4}{9}\right) = \frac{100}{243}$.
Correct Answer: 4