JEE Main 2023 · 6 April · evening · Single correct · +4 / −1
Three dice are rolled. If the probability of getting different numbers on the three dice is p/q, where p and q are co-prime, then q-p is equal to :
- (a) 3
- (b) 4
- (c) 1
- (d) 2
Answer and solution
Answer (b) 4
Solution
Maths sir · 3 steps
Fill the three dice one by one: 6 choices, then 5, then 4, out of 216 outcomes in all.
- Total outcomes. Each die shows one of 6 faces, and the dice are different objects. So there are 6 × 6 × 6 = 216 equally likely outcomes.
- Favourable outcomes. The first die can show any of 6 numbers. The second must avoid that number, so it has 5 choices. The third must avoid both, so it has 4. That gives 6 × 5 × 4 = 120 outcomes.
P = (6 × 5 × 4)/6³ = 120/216 = 5/9
- So. In lowest terms p = 5 and q = 9, which are co-prime. So q - p = 9 - 5 = 4, option (b).
The idea to keep. For all different numbers on n dice, count ordered outcomes in both places: 6 × 5 × 4 … on top, 6ⁿ below. Then reduce the fraction fully before reading off p and q.
Answer from the NTA final answer key. Solution by PrepWiser.