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 question, let's consider the possible outcomes for each letter.
When inserting the letters randomly, there are two possibilities for each letter:
1. It goes into the correct envelope.
2. It goes into an incorrect envelope.
Now, let's analyze the different scenarios for exactly three letters going into the correct envelopes:
Scenario 1: Only the first letter goes into the correct envelope. In this case, the second, third, and fourth letters would go into incorrect envelopes. The probability of this happening is (1/4) * (1/3) * (1/2) * (1/1) = 1/24.
Scenario 2: Only the second letter goes into the correct envelope. In this case, the first, third, and fourth letters would go into incorrect envelopes. The probability of this happening is (1/4) * (1/3) * (1/2) * (1/1) = 1/24.
Scenario 3: Only the third letter goes into the correct envelope. In this case, the first, second, and fourth letters would go into incorrect envelopes. The probability of this happening is (1/4) * (1/3) * (1/2) * (1/1) = 1/24.
Scenario 4: Only the fourth letter goes into the correct envelope. In this case, the first, second, and third letters would go into incorrect envelopes. The probability of this happening is (1/4) * (1/3) * (1/2) * (1/1) = 1/24.
Now, we need to sum up the probabilities of these four scenarios because we want exactly three letters to go into the correct envelopes.
Probability = Scenario 1 + Scenario 2 + Scenario 3 + Scenario 4 = 1/24 + 1/24 + 1/24 + 1/24 = 1/6.
Therefore, the correct answer is 1/6, which is not listed among the options provided.