03/17/2021 (2011: Countdown Round, Problem 22)

Q: When the letters ABCDE are randomly arranged, what is the probability that the two vowels are not next to each other? Express your answer as a common fraction.

A: 3/5

First, we find the probability that the vowels do end up next to each other. To do this, we treat "AE" as one unit, and find the number of ways that "AE", "B", "C", and "D" can be arranged. We get 4! = 24. Next, we multiply this by 2, because A and E can be arranged as "AE" or "EA". 2 ⋅ 24 = 48, which is the number of ways that ABCDE could be rearranged to have A and E next to each other.

Next, we find the total number of ways that ABCDE can be arranged, which is 5! = 120.

We find the probability that ABCDE can be arranges with the two vowels next to each other to be 48/120 = 2/5.

Lastly, we subtract this probability from 1 to get 3/5 as the probability that A and E are not next to each other.