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
$3$

Question

Three dice are rolled. If the probability of getting different numbers on the three dice is pq, where p and q are co-prime, then q - p is equal to:

Accepted Answer

Given: Three dice are rolled, and the probability of getting different numbers on the three dice is $\frac{p}{q}$ , where $p$ and $q$ are co-prime.

Find: The value of $q-p$ .

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 = 9-5 = 4

Therefore, the correct option is C.

Answer