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, 24 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-exclusion. The key is to find the total number of people who attended Dr. Rhonda's presentation, which involves accounting for those who saw combinations of the three movies: A, B, and C.

### Given dataTo solve this problem, we can use the principle of inclusion-exclusion. The key is to find the total number of people who attended Dr. Rhonda's presentation, which involves accounting for those who saw combinations of the three movies: A, B, and C.

### Given 
- \( |A| = 223 \) (people who saw movie A)
- \( |B| = 219 \) (people who saw movie B)
- \( |C| = 229 \) (people who saw movie C)
- \( |A \cap B| = 114 \) (people who saw both A and B)
- \( |A \cap C| = 121 \) (people who saw both A and C)
- \( |B \cap C| = 116 \) (people who saw both B and C)
- \( |A \cap B \cap C| = 24 \) (people who saw all three movies)
- 21 people saw none of the three movies.

### Applying the principle of inclusion-exclusion

The formula to find the total number of people who saw at least one movie is:

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

Substitute the given values into the formula:

\[
|A \cup B \cup C| = 223 + 219 + 229 - 114 - 121 - 116 + 24
\]

First, perform the additions and subtractions step-by-step:

\[
223 + 219 + 229 = 671
\]
\[
114 + 121 + 116 = 351
\]
\[
|A \cup B \cup C| = 671 - 351 + 24 = 344
\]

Thus, 344 people saw at least one of the three movies.

### Total number of attendees

Since 21 people saw none of the movies, the total number of people who attended the presentation is:

\[
\text{Total attendees} = 344 + 21 = 365
\]

Therefore, **365 people attended Dr. Rhonda's presentation**.

Would you like more details or have any questions? Here are some related questions to explore:

1. How many people saw only movie A?
2. How many people saw only movie B?
3. How many people saw only movie C?
4. How many people saw exactly two movies?
5. How would the result change if 30 people saw none of the movies?

**Tip**: When solving inclusion-exclusion problems, carefully distinguish between overlapping groups to avoid over-counting.

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, 24 had seen all three, and 21 had seen none of the three. How many people attended her presentation?

Learn how to solve an inclusion-exclusion problem involving the number of people who watched movies A, B, and C. Using set theory and the inclusion-exclusion principle, this detailed solution shows how to calculate the total number of attendees. Suitable for students in Grades 10-12.

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