Dr. Rhonda gave a presentation at a conference. She asked the audience whether they had seen movies A, B, and C, and gathered the following numbers:

223 people had seen A, 219 had seen B, 229 had seen C, 114 had seen A and B, 121 had seen A and C, 116 had seen B and C, 19 had seen all three, and 21 had seen none of the three.

How many people attended her presentation?

To solve this problem, we can use the principle of inclusion and exclusion in set theory. Let's define the following:

- \( |A| \) is the number of people who have seen movie A.
- \( |B| \) is the number of people who have seen movie B.
- \( |C| \) is the number of people who have seen movie C.
- \( |A \cap B| \) is the number of people who have seen both movies A and B.
- \( |A \cap C| \) is the number of people who have seen both movies A and C.
- \( |B \cap C| \) is the number of people who have seen both movies B and C.
- \( |A \cap B \cap C| \) is the number of people who have seen all three movies.

### Given values:

- \( |A| = 223 \)
- \( |B| = 219 \)
- \( |C| = 229 \)
- \( |A \cap B| = 114 \)
- \( |A \cap C| = 121 \)
- \( |B \cap C| = 116 \)
- \( |A \cap B \cap C| = 19 \)
- 21 people have seen none of the movies.

We are asked to find the total number of people who attended the presentation. The formula for the union of three sets is:

\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|
\]

Substituting the values:

\[
|A \cup B \cup C| = 223 + 219 + 229 - 114 - 121 - 116 + 19
\]

Let's calculate this.The total number of people who have seen at least one of the movies is 339. Since 21 people attended the presentation but did not see any of the three movies, the total number of attendees is:

\[
339 + 21 = 360
\]

Thus, 360 people attended Dr. Rhonda's presentation.

Would you like more details or have any questions? 

Here are 5 related questions you might explore:
1. What is the formula for inclusion-exclusion in set theory for four sets?
2. How would the calculation change if an additional movie D was introduced?
3. Can you determine the exact number of people who have seen only one of the three movies?
4. What percentage of attendees had seen all three movies?
5. How would you model this scenario using a Venn diagram?

**Tip:** When dealing with multiple sets, visualize the overlap with Venn diagrams to make the inclusion-exclusion principle clearer.

Dr. Rhonda gave a presentation at a conference. She asked the audience whether they had seen movies A, B, and C, and gathered the following numbers: 223 people had seen A, 219 had seen B, 229 had seen C, 114 had seen A and B, 121 had seen A and C, 116 had seen B and C, 19 had seen all three, and 21 had seen none of the three. How many people attended her presentation?

Discover how to solve a set theory problem using the inclusion-exclusion principle. Dr. Rhonda gathered data on 223 people who had seen movie A, 219 had seen B, and 229 had seen C, with overlaps of 114, 121, and 116 people. Learn how the inclusion-exclusion formula helps to calculate the total number of attendees. Suitable for Grades 10-12.

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