While the given answer does yield the correct result, I don't see how it is intuitive that the probability of having at least one correct in the possibilities that remain is 3/6. There are 18 permutations remaining and 9 of them have at least one correct, but I don't see how that clearly gives us 3/6.

As for possible answers: 1.) I answered the question by listing all permutations and identifying the permutations with at least one correct. (small-brain but time-consuming)

2.) AI used a formula for derangements. (annoying formula to memorize)

Neither of these methods are very satisfying so I didn't make an edit

I believe you are referring to this exercise.

First approach (fancy)

1. Label the envelopes 1,2,3,4 and the letters 1,2,3,4.
2. If envelope 1 doesn't contain letter 1 then surely another envelope won't have the right letter (the one containing letter 1)
3. Hence any matching pair of envelope-letter must occur with 2 envelopes (among 3 remaining) not equal to envelope 1 and 2 letters (among the 3 remaining) not equal to letter 1
4. If you choose two numbers from {2,3,4} and again two numbers from {2,3,4}, one of the numbers from the first pair matches one of the numbers from the second pair
5. Hence given that we are left with 2 envelopes not equal to envelope 1 and 2 letters not equal to letter 1, one of the two remaining letters must match with one of the two remaining envelopes, and trivially a match is possible with probability 1/2

Hence the total probability is 1/4 + 3/4 * 1/2 = 5/8.

Second approach (hack)

Now I will give you a simpler way of doing this. Let's say we want to calculate the probability of a matching arrangement given that the first envelope does not contain letter 1. Then because letters are assigned randomly, the probability of a matching arrangement does not depend on which (wrong) letter is contained in envelope 1, and similarly it should not depend on which envelope contains letter 1. But the conditional probability of a matching arrangement given that envelope 1 contains letter 2 and envelope 2 contains letter 1 is obviously 1/2.

Third approach (computational/AI)

You can do this problem using the inclusion-exclusion formula with 4 events where you look at events with 1 match, 2 matches, 3 matches, 4 matches.

AI systems might tell you that the first two approaches are wrong but they are in fact perfectly sound. In fact, I was able to convince Gemini that both solutions are valid after some explanation and prodding.