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Let's solve the problem step by step.

### Given Information:
- Total number of students: 20
- Students wearing a T-shirt but not shorts: 8
- Students wearing a T-shirt: 12
- Students wearing neither a T-shirt nor shorts: 5

### Steps:

1. **Let**:
   - \( T \) be the set of students wearing a T-shirt.
   - \( S \) be the set of students wearing shorts.
   
2. **Identify the given quantities**:
   - \( |T \cap S'| = 8 \) (Students wearing only a T-shirt)
   - \( |T| = 12 \) (Students wearing a T-shirt, including those wearing only T-shirt and those wearing both)
   - \( |T' \cap S'| = 5 \) (Students wearing neither a T-shirt nor shorts)

3. **Calculate**:
   - Students who wear either T-shirt or shorts (or both):
     \[
     |T \cup S| = \text{Total number of students} - \text{Students wearing neither} = 20 - 5 = 15
     \]
   - Students who wear shorts, whether they wear T-shirts or not:
     \[
     |S| = |T \cup S| - |T| + |T \cap S'|
     = 15 - 12 + 8
     = 11
     \]
   - Students who wear both T-shirt and shorts:
     \[
     |T \cap S| = |T| - |T \cap S'| = 12 - 8 = 4
     \]
   - Students wearing only shorts:
     \[
     |S \cap T'| = |S| - |T \cap S| = 11 - 4 = 7
     \]

### **Venn Diagram Analysis**:
- **T-shirt Only (A)** = 8
- **Both T-shirt and Shorts (B)** = 4
- **Shorts Only (C)** = 7
- **Neither (D)** = 5

### Final Answer:
- Number of students wearing shorts (which includes those wearing only shorts and those wearing both): \( 4 + 7 = 11 \).

Would you like me to create a Venn diagram to illustrate this? 

### Follow-Up Questions:
1. What would happen if no students were wearing neither a T-shirt nor shorts?
2. If one more student wore only shorts, how would that change the solution?
3. How can you calculate the number of students wearing only T-shirts if the total number changes?
4. What does the result say about the climate or conditions implied by the data?
5. Can we solve the problem with different initial conditions, such as a different number of students?

### Tip:
When working with sets and Venn diagrams, always double-check that the sums of individual groups match the total population.

In this hot weather, a class of 20 students has 8 students who are wearing a t-shirt but not shorts, 12 wearing a t-shirt, 5 that are wearing neither a t-shirt nor shorts. Draw a Venn diagram to show the above information, then solve the diagram. Finally, answer the question: How many of the students were wearing shorts?

Solve a Venn diagram problem involving a class of 20 students, with a focus on students wearing t-shirts, shorts, or neither. This problem covers key concepts like set theory, Venn diagrams, and the principle of inclusion-exclusion, suitable for Grades 9-10.

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