Key Concepts and Formulas

• Probability Definition: For an event $E$ in a sample space of equally likely outcomes, its probability is $P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$.
• Fundamental Principle of Counting (Multiplication Principle): If an event can occur in $m$ ways and another independent event can occur in $n$ ways, then both events can occur in $m \times n$ ways.
• Combinations ($^nC_r$): The number of ways to choose $r$ distinct items from a set of $n$ distinct items, where the order of selection does not matter, is $^nC_r = \frac{n!}{r!(n-r)!}$.
• Distributing Distinct Items to Distinct Bins (Surjective Mapping): The number of ways to distribute $n$ distinct items into $k$ distinct bins such that exactly $r$ specified bins are non-empty is given by $r! S(n,r)$, where $S(n,r)$ is a Stirling number of the second kind. For $k=2$, this simplifies to $2^n - 2$.

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Step-by-Step Solution

Step 1: Calculate the Total Number of Possible Outcomes

• What we are doing: Determining all possible ways to post three distinct letters to five distinct addresses. This forms our sample space.

• Why this approach: Each of the three letters can be posted to any one of the five addresses, independently. Since both letters and addresses are distinct, we use the multiplication principle.

• Calculation:

  • Letter 1 has 5 choices of addresses.
  • Letter 2 has 5 choices of addresses.
  • Letter 3 has 5 choices of addresses.

  Therefore, the total number of ways to post the 3 letters is: $\text{Total Outcomes} = 5 \times 5 \times 5 = 5^3 = 125$

Step 2: Calculate the Number of Favorable Outcomes (Letters posted to exactly two addresses)

• What we are doing: We need to find the number of ways such that exactly two of the five addresses receive at least one letter. This is a two-stage process: first, choose the two addresses, and then distribute the letters to them.

Stage 2a: Select the two specific addresses out of five that will receive letters.

• Why this approach: The order in which we choose the two addresses does not matter (e.g., choosing Address A then Address B is the same as choosing Address B then Address A). This calls for combinations.
• Calculation: The number of ways to select 2 addresses from 5 is: ${^5C_2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$

Stage 2b: Distribute the 3 distinct letters to the chosen 2 addresses such that both addresses receive at least one letter.

• What we are doing: Let's say we have chosen two specific addresses, $A_X$ and $A_Y$. We need to post the 3 distinct letters (L1, L2, L3) to these two addresses such that neither $A_X$ nor $A_Y$ is left empty.
• Why this approach (using Inclusion-Exclusion for surjective functions):
  1. First, consider all possible ways to post the 3 distinct letters to these 2 distinct addresses ($A_X$ and $A_Y$). Each letter has 2 choices, so there are $2^3 = 8$ total ways.
  2. However, this count includes cases where only one of the chosen addresses is used. We must subtract these invalid cases:
     • All 3 letters go to Address $A_X$ (1 way).
     • All 3 letters go to Address $A_Y$ (1 way).
  3. Subtracting these cases ensures that both addresses receive at least one letter.
• Calculation: Number of ways to post 3 letters to exactly 2 chosen addresses $= (\text{Total ways to post to 2 addresses}) - (\text{Ways to post to only 1 of the 2 addresses})$ $= 2^3 - 2 = 8 - 2 = 6$ This is also equivalent to $2! S(3,2) = 2 \times 3 = 6$.

Stage 2c: Combine the two stages for total favorable outcomes.

• What we are doing: For each way of choosing 2 addresses (from Stage 2a), there are a certain number of ways to distribute the letters such that both selected addresses are used (from Stage 2b). We multiply these possibilities.
• Calculation: Number of Favorable Outcomes = (Ways to choose 2 addresses) $\times$ (Ways to distribute 3 letters to these 2 addresses using both) $= {^5C_2} \times 6 = 10 \times 6 = 60$

Step 3: Calculate the Probability

• What we are doing: Apply the probability definition using the total outcomes from Step 1 and favorable outcomes from Step 2.
• Calculation: $P(\text{exactly two addresses}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{60}{125}$
• Simplifying the Fraction: Divide both the numerator and denominator by their greatest common divisor, which is 5. $\frac{60}{125} = \frac{60 \div 5}{125 \div 5} = \frac{12}{25}$

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Common Mistakes & Tips

• "Exactly two addresses" vs. "at least two" or "at most two": Pay close attention to the wording. "Exactly two" means the number of addresses receiving letters must be precisely two, no more, no less.
• Distinct vs. Identical Items/Bins: In this problem, both letters and addresses are distinct. If they were identical, different combinatorial formulas (e.g., stars and bars) would be required.
• The "Onto" Condition: When distributing items to a selected set of bins such that all bins in that set must be used, remember to subtract cases where some bins are left empty. A common error is to simply use $k^n$ for the distribution without accounting for the "exactly" constraint.
• Systematic Breakdown: Break down complex probability problems into smaller, manageable steps: calculate total outcomes, then favorable outcomes (often requiring sub-steps like selection and then distribution), and finally, calculate the probability.

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Summary

The problem requires calculating the probability that three distinct letters are posted to exactly two out of five distinct addresses. We first determine the total number of ways to post the letters, which is $5^3 = 125$. Then, we calculate the number of favorable outcomes by first choosing the two addresses (${^5C_2} = 10$ ways) and then distributing the three letters to these two chosen addresses such that both receive at least one letter ($2^3 - 2 = 6$ ways). Multiplying these gives $10 \times 6 = 60$ favorable outcomes. Finally, the probability is $\frac{60}{125}$, which simplifies to $\frac{12}{25}$.

The final answer is $\boxed{\frac{12}{25}}$, which corresponds to option (B).