Jack rolls 5 fair six-sided dice. What is the probability that at least two dice show the same number?
The only way in which there won't be two dice displaying the same number is if exactly one number between 1 and 6 isn't being displayed on any of the dice and the 5 dice all display different numbers. There are 6 different possibilities for the number that isn't being displayed, and then there are a total of $5!$ ways in which the 5 dice displaying different numbers can be arranged, so there are a total of $6\cdot5!$ outcomes that result in the dice all displaying different numbers. Since each of the 5 dice can have 6 results when rolled and all of the rolls are determined independently, there are a total of $6^5$ possible outcomes, which means that the probability of all of the dice showing different numbers is $\dfrac{6\cdot5!}{6^5}=\dfrac{5}{54}$, so the probability that we want is $1-\dfrac{5}{54}=\boxed{\dfrac{49}{54}}$.