GATE CS 2024 Set 2 — Question 18

When rolling 6 dice, the total number of possible outcomes is $6^6$.For the numbers to be all distinct (i.e., getting exactly one 1, one 2, ..., one 6), we are looking for the number of permutations of the set $\{1, 2, 3, 4, 5, 6\}$.
The number of favorable outcomes is $6!$ (6 factorial).The probability $P$ is:$$P = \frac{6!}{6^6} = \frac{720}{46656}$$Simplifying the fraction:$$P = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 6 \times 6 \times 6 \times 6 \times 6} = \frac{5 \times 4 \times 3 \times 2 \times 1}{6^5}$$$$P = \frac{120}{7776}$$Dividing numerator and denominator by 24:$$120 \div 24 = 5$$$$7776 \div 24 = 324$$Thus, $P = \frac{5}{324}$.