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 find the probability of exactly three letters going into the correct envelopes, we need to consider the total number of possible arrangements and the number of favorable outcomes.
Let's break down the problem step by step:
Step 1: Total number of possible arrangements
Since there are four letters and four envelopes, there are 4! (read as 4 factorial) ways to arrange the letters in the envelopes. The factorial of a number is the product of all positive integers from that number down to 1. So, 4! = 4 x 3 x 2 x 1 = 24.
Step 2: Number of favorable outcomes
For exactly three letters to go into the correct envelopes, one letter must be placed randomly. Let's say we choose letter A and put it in its correct envelope. Now, we have three more letters (B, C, and D) that need to be placed in their respective envelopes.
There are two possibilities for each of the remaining letters:
1) Letter B can either be placed correctly or incorrectly.
2) Letter C can either be placed correctly or incorrectly.
3) Letter D will automatically be placed correctly since there is only one envelope left unassigned.
So, the total number of ways to arrange the remaining three letters while fixing one correct placement (A) is 2 x 2 x 1 = 4.
Step 3: Calculate the probability
The probability is the ratio of favorable outcomes to total possible outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
= 4 / 24
= 1 / 6
Therefore, the correct answer is 1/6, which is not one of the options provided. None of the given options (1/2, 1/3, 1/4, 0) is the correct probability for exactly three letters going into the correct envelopes.