What is the probability that exactly one letter out of four is placed into its correct envelope when randomly distributing them?

To determine the probability that exactly one letter out of four is placed into its correct envelope when randomly distributing them, we can analyze the problem using combinatorial principles.

When one letter is correctly placed, the other three letters must all be placed incorrectly. This scenario can be examined through the concept of derangements, which refers to the number of ways to arrange a set of items so that none of the items appear in their original position.

For four letters, we denote the total number of permutations as (4!) (which equals 24), since there are four letters that can be arranged in every possible order.

Next, we need to calculate the number of ways to achieve exactly one letter being placed correctly. First, we can select which one of the four letters is the correctly placed one, which can occur in four different ways. After choosing one letter to be positioned correctly, we are left with three letters that need to be deranged (none should end up in the correct envelope).

The number of derangements (denoted as (D_n)) of (n) items can be calculated using the formula:

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D_n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i