Question

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:

• A. 1
• B. 2
• C. 4
• D. 3

Answer 1

The Correct Option is C

The number of favorable outcomes where the three dice show different numbers is: \[ \binom{6}{3} \times 3! = 20 \times 6 = 120 \] The total number of possible outcomes when rolling three dice is: \[ 6 \times 6 \times 6 = 216 \] Thus, the probability is: \[ P = \frac{120}{216} = \frac{5}{9} \] So, \(p = 5\) and \(q = 9\), and \(q - p = 4\).