Probability
Probability
star_batch_jee_advanced_2025
Grade 12
Question:
Three different dice are rolled three times. The Probability that they show different numbers only two times is:
1. $\frac{1}{3}$
2. $\left(\frac{6}{6^3}\right)^2$
3. $\frac{107}{(54)^3}$
4. $\frac{100}{243}$
Step-by-Step Solution
Key Concept: Apply the permutation formula for all different outcomes, then use conditional probability to find cases with exactly one matching pair.
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