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### Given Information:
- **Total professional athletes surveyed** = 251
- **Owned a convertible** = 108
- **Owned a giant screen TV** = 107
- **Owned a sporting goods store** = 112
- **Owned both a convertible and a store** = 19
- **Owned both a TV and a store** = 50
- **Owned both a convertible and a TV** = 46
- **Owned all three (convertible, TV, store)** = 11
- **Athletes who did not own any of the three items** = 28

### Questions to be Answered:
1. **How many athletes did not own any of the three items?**
   - Answer: **28** (already provided in the problem).

2. **How many owned a convertible and a TV, but not a store?**
   - The number of athletes who owned both a convertible and a TV is 46. However, 11 of these owned all three (including the store). Therefore, the number who owned a convertible and a TV but not a store is:
     \[
     46 - 11 = 35
     \]
     **Answer: 35**

3. **How many athletes owned a convertible or a TV?**
   - To calculate this, we can apply the inclusion-exclusion principle. Let:
     - \( C \) = number of athletes who own a convertible = 108
     - \( T \) = number of athletes who own a TV = 107
     - \( C \cap T \) = number of athletes who own both a convertible and a TV = 46

     The total number who own either a convertible or a TV (or both) is:
     \[
     |C \cup T| = C + T - (C \cap T) = 108 + 107 - 46 = 169
     \]
     **Answer: 169**

4. **How many athletes owned exactly one type of item in the survey?**
   - We need to subtract those who owned more than one type from the total athletes who own at least one type of item.
   - Total owning at least one item = Total surveyed - athletes with none = 251 - 28 = 223.
   - From this, subtract the number of athletes who owned more than one item (i.e., combinations like convertible and store, TV and store, etc.):
     \[
     19 \, (\text{convertible and store}) + 50 \, (\text{TV and store}) + 46 \, (\text{convertible and TV}) - 2 \times 11 \, (\text{all three to correct double-counting}) = 104
     \]
     Therefore, athletes owning exactly one item:
     \[
     223 - 104 = 119
     \]
     **Answer: 119**

5. **How many athletes owned at least one type of item in the survey?**
   - This is the complement of the athletes who owned none:
     \[
     251 - 28 = 223
     \]
     **Answer: 223**

6. **How many owned a TV or a store, but not a convertible?**
   - From the inclusion-exclusion principle again, for TV and store (without a convertible), we can calculate this by subtracting those who owned a convertible from the total who owned either a TV or store:
     \[
     (107 + 112 - 50) - (46 - 11) = \text{needs more detailed breakdown}
     \]

Would you like a more detailed breakdown of the last question or further explanations on any part of the solution?

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Here are 5 related questions:
1. How would you find how many athletes owned exactly two of the items?
2. Can you determine how many athletes owned only a convertible?
3. How many owned only a TV and not the other two items?
4. What is the minimum number of athletes who owned a store?
5. Can we determine how many athletes owned exactly a store and nothing else?

**Tip:** When solving problems involving multiple sets and their intersections, the inclusion-exclusion principle is very useful!

In a survey of 251 professional athletes, it was found that 108 of them owned a convertible, 107 of them owned a giant screen TV, and 112 owned a sporting goods store. 19 owned a convertible and a store, 50 owned a TV and a store, and 46 owned a convertible and a TV. 11 owned all three items. How many athletes owned none, exactly one, or at least one of these items?

Explore a detailed solution to a set theory problem involving 251 athletes and their ownership of convertibles, TVs, and stores. Learn to apply the inclusion-exclusion principle and calculate athletes owning none, exactly one, or at least one item. Suitable for high school students.

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