In a room, there are four people - Alice, Bob, Carol, and David. They each randomly choose a number between 1 and 4, inclusive. What is the probability that no two people choose the same number?
The probability is 1/6.

Here's the reasoning:
1. There are a total of 4! (4 factorial) ways for the four people to choose their numbers, which is equal to 4 * 3 * 2 * 1 = 24.
2. For no two people to choose the same number, each person must choose a different number from the others.
3. There are also 4! ways for this to happen since there are 4 choices for Alice, 3 remaining choices for Bob, 2 remaining choices for Carol, and only 1 choice left for David. This gives us another 4 * 3 * 2 * 1 = 24 possible arrangements where no two people choose the same number.
4. Therefore, the probability that no two people choose the same number is the number of successful outcomes (no repeats) divided by the total number of possible outcomes: 24/24 = 1/6.