Three dice are rolled. If the probability of getting different numbers on the three dice is where and are co-prime, then is equal to [2023]

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Solution

Q.
Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q},$ where $p$ and $q$ are co-prime, then $q - p$ is equal to [2023]

1 4

2 3

3 1

4 2

Ans.
(1)

If numbers are different on all three dice,

Total number of favourable outcomes $= 6 \times 5 \times 4 = 120$

Total number of possible outcomes $= 6^{3} = 216$

$∴$ $Required probability = \frac{120}{216} = \frac{5}{9} = \frac{p}{q}$

So, $q - p = 9 - 5 = 4$

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