GATE CS 2024 Set 2 Q8 — Probability: Random Variables and Distributions

To find the probability of getting all distinct numbers when six unbiased dice are rolled simultaneously, we calculate the ratio of favorable outcomes to the total possible outcomes.

1. Total Number of Outcomes:
Each of the $6$ dice has $6$ possible faces. When $6$ dice are rolled, the total number of outcomes is: $$\text{Total Outcomes} = 6^6$$

2. Number of Favorable Outcomes:
For all $6$ dice to show distinct numbers, the outcomes must be a permutation of the set $\{1, 2, 3, 4, 5, 6\}$.
The number of ways to arrange these $6$ distinct numbers is: $$\text{Favorable Outcomes} = 6! = 720$$

3. Probability Calculation:
$$\text{Probability } (P) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6!}{6^6}$$ Simplifying the expression: $$P = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 6^5} = \frac{120}{6^5} = \frac{120}{7776}$$ Dividing both the numerator and the denominator by $24$: $$P = \frac{120 \div 24}{7776 \div 24} = \frac{5}{324}$$

Thus, the correct option is B.