Thirty-one cities were researched to determine whether they had a professional sports​ team, a​ symphony, or a​ children's museum. Of these​ cities, 17 had a professional sports​ team, 17 had a​ symphony, 11 had a​ children's museum, 13 had a professional sports team and a​ symphony, 5 had a professional sports team and a​ children's museum, 6 had a symphony and a​ children's museum, and 4 had all three activities. How many of the cities surveyed had a professional sports team or a​ symphony?

To solve this problem, we'll use the principle of inclusion-exclusion (PIE). Here's how we can approach it:

Let:
- \( A \) be the set of cities with a professional sports team.
- \( B \) be the set of cities with a symphony.
- \( C \) be the set of cities with a children's museum.

Given:
- \( |A| = 17 \) (cities with a professional sports team)
- \( |B| = 17 \) (cities with a symphony)
- \( |C| = 11 \) (cities with a children's museum)
- \( |A \cap B| = 13 \) (cities with both a professional sports team and a symphony)
- \( |A \cap C| = 5 \) (cities with both a professional sports team and a children's museum)
- \( |B \cap C| = 6 \) (cities with both a symphony and a children's museum)
- \( |A \cap B \cap C| = 4 \) (cities with all three activities)

We are asked to find the number of cities that had a professional sports team or a symphony, i.e., \( |A \cup B| \).

Using the principle of inclusion-exclusion (PIE):

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

However, since there is also a third category (children's museum), and we are not considering it directly for this specific question, we don't need to subtract any intersections with set \( C \).

Substitute the given values:

\[
|A \cup B| = 17 + 17 - 13 = 21
\]

So, **21 cities** had either a professional sports team or a symphony.

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

1. How many cities had a professional sports team or a children's museum?
2. How many cities had only one of the three activities?
3. How many cities had none of the three activities?
4. How many cities had exactly two of the three activities?
5. What is the total number of cities that had any of the three activities?

**Tip:** The principle of inclusion-exclusion (PIE) is a powerful tool for counting the number of elements in the union of overlapping sets.

Learn how to solve a problem involving 31 cities with professional sports teams, symphonies, and children's museums using the principle of inclusion-exclusion. Discover how many cities had either a professional sports team or a symphony.

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