Thirtydash three cities were researched to determine whether they had a professional sports​ team, a​ symphony, or a​ children's museum. Of these​ cities, 14 had a professional sports​ team, 16 had a​ symphony, 14 had a​ children's museum, 9 had a professional sports team and a​ symphony, 7 had a professional sports team and a​ children's museum, 8 had a symphony and a​ children's museum, and 4 had all three activities. Complete parts ​a) through ​e) below. ) How many of the cities surveyed had a professional sports team and a​ symphony, but not a​ children's museum?

To solve this problem, we can use the principle of inclusion-exclusion (PIE) and a Venn diagram to keep track of the different sets of cities.

### Definitions:
- Let \( P \) represent the set of cities with a professional sports team.
- Let \( S \) represent the set of cities with a symphony.
- Let \( M \) represent the set of cities with a children's museum.

### Given Data:
- \( |P| = 14 \) (cities with a professional sports team)
- \( |S| = 16 \) (cities with a symphony)
- \( |M| = 14 \) (cities with a children's museum)
- \( |P \cap S| = 9 \) (cities with a professional sports team and a symphony)
- \( |P \cap M| = 7 \) (cities with a professional sports team and a children's museum)
- \( |S \cap M| = 8 \) (cities with a symphony and a children's museum)
- \( |P \cap S \cap M| = 4 \) (cities with all three)

### Problem: 
We want to find the number of cities that had a professional sports team and a symphony, but **not** a children's museum, i.e., \( |P \cap S \cap \overline{M}| \).

### Solution Steps:

1. **Calculate \( |P \cap S| \):** 
   - We know that \( |P \cap S| = 9 \).

2. **Subtract the cities that have all three activities \( |P \cap S \cap M| \):**
   - The cities that have a professional sports team, a symphony, and a children's museum are \( |P \cap S \cap M| = 4 \).

3. **Find \( |P \cap S \cap \overline{M}| \):**
   - These are the cities with a professional sports team and a symphony, but **not** a children's museum. So, subtract \( |P \cap S \cap M| \) from \( |P \cap S| \):
   \[
   |P \cap S \cap \overline{M}| = |P \cap S| - |P \cap S \cap M|
   \]
   \[
   |P \cap S \cap \overline{M}| = 9 - 4 = 5
   \]

### Conclusion:
The number of cities that had a professional sports team and a symphony but not a children's museum is **5**.

Would you like more details or have any questions?

Here are some related questions:
1. How many cities had only a professional sports team?
2. How many cities had only a symphony?
3. How many cities had only a children's museum?
4. How many cities had exactly two of the three activities?
5. What is the total number of cities surveyed?

**Tip:** The principle of inclusion-exclusion is a useful tool for counting in overlapping sets, especially when dealing with multiple categories.

Learn how to solve a problem involving cities with professional sports teams, symphonies, and children's museums using the principle of inclusion-exclusion and Venn diagrams. Explore the intersection of sets and count cities based on given data. Suitable for high school students.

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