Can you calculate the number of students who play exactly two sports

To calculate the number of students who play exactly two sports, we need to determine the values for students who play exactly two sports in each combination:

1. **Basketball and Football but not Tennis**: \( a \)
2. **Basketball and Tennis but not Football**: \( b \)
3. **Football and Tennis but not Basketball**: \( c \)

We already know from the previous breakdown:

- \( a + b = 3 \)
- \( a + c = 4 \)
- \( b + c = 2 \)
- \( d = 7 \) (students who play all three sports)

We can find each variable as follows:

### Step 1: Solving for \( a \), \( b \), and \( c \)
- From \( a + b = 3 \) and \( b + c = 2 \), subtract the second equation from the first:
  \[
  (a + b) - (b + c) = 3 - 2 \Rightarrow a - c = 1 \quad \text{(Equation 1)}
  \]
- From \( a + c = 4 \) and \( a - c = 1 \) (from Equation 1), add the equations:
  \[
  (a + c) + (a - c) = 4 + 1 \Rightarrow 2a = 5 \Rightarrow a = 2.5 \quad \text{(which is not possible in a real-world scenario, meaning there might be a typo or misinterpretation of data)}.
  \]

Given this issue, let's try using logical steps with integer constraints.

### Step 2: Consider that \( d = 7 \) and recheck the sums
- \( a + b + 7 = 10 \Rightarrow a + b = 3 \)
- \( a + c + 7 = 11 \Rightarrow a + c = 4 \)
- \( b + c + 7 = 9 \Rightarrow b + c = 2 \)

If \( d = 7 \), we can:

### Recalculate \( a \), \( b \), \( c \)
- \( a = 2 \), \( b = 1 \), \( c = 0 \) (These are guesses based on the sums, but trying the only whole-number combination):
  - \( a + b = 2 + 1 = 3 \) (Correct)
  - \( a + c = 2 + 0 = 2 \) (Seems incorrect, but keeping to realistic integer answers)

### Result:
- **Exact** \( a + b + c = 2 + 1 + 0 = 3 \), so the exact solution might have been affected by data entry.

#### Answer:
- Students playing exactly two sports: \( 3 + 0 + 2 \), this is an error. Please double-check with the full Venn details to correct for potential misreading.

Let me know if you want an adjusted analysis or rechecking for this issue! 

### Related Questions:
1. What if an additional sport was added to the Venn?
2. How would a bar chart represent the three-set overlaps?
3. Can you compare the intersection sizes to a four-sport solution?
4. How would adding exclusions to any sport affect results?
5. Can you visually interpret these results with pie-charts?

### Tip:
Always analyze both equation sums and data sets for mismatches that lead to non-integer or unrealistic solutions!

Learn how to calculate the number of students who play exactly two sports using a detailed breakdown of Venn diagram intersections and a system of equations approach. Understand the challenges in data interpretation and find realistic integer solutions. Suitable for high school students studying set theory and advanced problem-solving techniques.

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