Thirty 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 only a professional sports​ team?

To find out how many cities had only a professional sports team (and not a symphony or a children's museum), we need to use the principle of inclusion-exclusion and the given data.

### Given Information:
- Total cities: \( n = 33 \)
- Cities with a professional sports team \( |P| = 14 \)
- Cities with a symphony \( |S| = 16 \)
- Cities with a children's museum \( |C| = 14 \)
- Cities with both a professional sports team and a symphony \( |P \cap S| = 9 \)
- Cities with both a professional sports team and a children's museum \( |P \cap C| = 7 \)
- Cities with both a symphony and a children's museum \( |S \cap C| = 8 \)
- Cities with all three activities \( |P \cap S \cap C| = 4 \)

### Finding Cities with Only a Professional Sports Team:

Cities with only a professional sports team are those that have a professional sports team but do not have a symphony or a children's museum.

To calculate this, we subtract the number of cities that have both a professional sports team and a symphony, both a professional sports team and a children's museum, and all three activities from the total number of cities with a professional sports team.

\[
|P \text{ only}| = |P| - |P \cap S| - |P \cap C| + |P \cap S \cap C|
\]

Substituting the values:

\[
|P \text{ only}| = 14 - 9 - 7 + 4 = 2
\]

### Answer:
**2 cities had only a professional sports team.**

Would you like further explanation on this calculation or have any other questions?

Here are some related questions to consider:
1. How many cities had only a symphony?
2. How many cities had only a children's museum?
3. How many cities had exactly two of the three activities?
4. How many cities had none of the activities?
5. How many cities had at least one of the three activities?

**Tip:** When calculating how many elements belong to only one set in problems involving multiple sets, always consider subtracting the elements that belong to the intersections with other sets.

Explore how to determine the number of cities with only a professional sports team among those surveyed, using the principle of inclusion-exclusion. This problem involves set theory and provides a step-by-step solution to find the cities meeting specific criteria.

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