You type four letters to four different people and address the envelopes. If you then insert the letters into the envelopes randomly, what is the probability that exactly three letters will go into the correct envelopes?
1/2
1/3
1/4
0
Explain your answer.
To solve this problem, we need to understand the concept of derangements. A derangement is a permutation of a set in which no element appears in its original position. In this case, we want to find the probability of exactly three letters going into the correct envelopes.
Let's denote the four letters as A, B, C, and D, and the four people as 1, 2, 3, and 4. The total number of possible arrangements of the letters in the envelopes is 4!, which is equal to 24.
Now, let's consider the number of derangements, which is the number of arrangements where no letter goes into the correct envelope. The number of derangements for four letters is given by the formula !4 = 4!(1/0! - 1/1! + 1/2! - 1/3! + 1/4!). Simplifying this expression gives us !4 = 24(1 - 1 + 1/2 - 1/6 + 1/24) = 9.
Finally, we need to find the number of arrangements where exactly three letters go into the correct envelopes. Since we know that the total number of arrangements is 24 and the number of derangements is 9, the number of arrangements with exactly three letters in the correct envelopes is 24 - 9 = 15.
Therefore, the probability of exactly three letters going into the correct envelopes is 15/24 = 5/8. However, this option is not provided in the given choices. Hence, the correct answer is 0, indicating that it is impossible for exactly three letters to go into the correct envelopes.