*calculae 3rd partue using inclusive exclusive principal*

To calculate the probability that exactly three letters are in the correct envelope using the Inclusion-Exclusion Principle, we need to follow these steps:

### Step 1: Define the Events
Let \(A_i\) represent the event that the \(i\)th letter is in the correct envelope. We are interested in finding the probability of exactly three of these events occurring.

### Step 2: Use the Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle is used to find the union of multiple sets. In this case, we want the union of exactly three of these sets \(A_1, A_2, A_3\), and so on. To use this principle, we calculate the following:

\[
P(\text{exactly 3 correct}) = \sum_{|S|=3} P\left(\bigcap_{i \in S} A_i \cap \bigcap_{j \notin S} A_j^c \right)
\]

where \(S\) is a subset of {1, 2, 3, 4, 5} with exactly 3 elements.

### Step 3: Calculate Probabilities
1. **Total ways to choose 3 letters to be correct:** There are \(\binom{5}{3} = 10\) ways to choose 3 letters out of 5 to be correct.

2. **Number of derangements of the remaining 2 letters:** If exactly 3 letters are correctly placed, the remaining 2 letters must be incorrectly placed. The number of ways to incorrectly place the remaining 2 letters (derangement of 2 elements) is:

\[
D_2 = 2! \left(\frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!}\right) = 2 \times \left(1 - 1 + \frac{1}{2}\right) = 2 \times \frac{1}{2} = 1
\]

3. **Calculate the total number of favorable outcomes:** For each of the \(\binom{5}{3}\) choices, there is 1 favorable derangement of the remaining 2 letters, so the total number of favorable outcomes is:

\[
10 \times 1 = 10
\]

4. **Calculate the probability:** Since there are \(5!\) total possible arrangements of the letters, the probability is:

\[
P(\text{exactly 3 correct}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{10}{120} = \frac{1}{12}
\]

### Final Answer
The probability that exactly three letters are in the correct envelope, using the Inclusion-Exclusion Principle, is \(\frac{1}{12}\).

Would you like more details on the Inclusion-Exclusion Principle or how it applies to other problems?

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**Related Questions:**

1. How would you calculate the probability for exactly two letters being in the correct envelope using the Inclusion-Exclusion Principle?
2. Can the Inclusion-Exclusion Principle be applied to events with more than three elements?
3. How does the Inclusion-Exclusion Principle relate to the formula for derangements?
4. What is the difference between using the Inclusion-Exclusion Principle and direct counting methods?
5. How can the Inclusion-Exclusion Principle be generalized for any number of elements?
6. Can you apply the Inclusion-Exclusion Principle to independent events?
7. What is the role of overlapping events in the Inclusion-Exclusion Principle?
8. How can the Inclusion-Exclusion Principle be used in real-life applications?

**Tip:** The Inclusion-Exclusion Principle is particularly useful in counting problems where you need to subtract over-counted cases to get the correct total.

Learn how to calculate the probability that exactly three letters are in the correct envelope using the Inclusion-Exclusion Principle. Understand the application of combinatorics and probability theory, including the binomial coefficient and derangement formula. Suitable for advanced high school or college students.

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