Question

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

• A. $\frac{12}{25}$
• B. $\frac{18}{25}$
• C. $\frac{4}{25}$
• D. $\frac{6}{25}$

Answer 1

The Correct Option is A

The objective is to determine the probability that precisely two distinct addresses are utilized when sending three letters to one of five available addresses.

1. First, compute the total number of possible assignments for posting three letters to any of the five addresses. Given that each letter's destination is independent, the total possibilities are \(5^3 = 125\).
2. Next, identify the number of favorable outcomes where exactly two addresses are selected.
3. Select 2 out of the 5 available addresses for the letters. The number of ways to do this is \(\binom{5}{2} = 10\).
4. After selecting the two addresses, distribute the 3 letters such that each selected address receives at least one letter. This distribution scenario, where exactly two addresses are used, can be categorized into two configurations:
   • Configuration 1: One address receives a single letter, and the other address receives the remaining two letters.
   • Configuration 2: The reverse of Configuration 1, where the addresses are swapped.
5. For Configuration 1: Choose 1 letter out of 3 to send to the first address (3 ways), and the remaining 2 letters are sent to the second address (1 way). This yields \(\binom{3}{1} \cdot 1 = 3\) ways.
6. Considering the symmetry, where either of the chosen addresses can receive the single letter or the pair of letters, an additional 3 ways are generated by swapping the letter assignments. Therefore, the total ways for distributing the letters to two specific addresses are \(3 + 3 = 6\).
7. For each pair of selected addresses, there are 6 distinct ways to distribute the three letters. Consequently, the total number of favorable outcomes is \(10 \cdot 6 = 60\).
8. Finally, the probability is calculated as the ratio of favorable outcomes to the total possible outcomes: \(\frac{60}{125} = \frac{12}{25}\).

Consequently, the probability that the three letters are posted to exactly two distinct addresses is \(\frac{12}{25}\).