There are 81 students in the Travel Club. They discovered that 34 members have visited  country​ A, 38 have visited country​ B, 33 have been to country​ C, 12 have visited countries A and​ B, 14 have been only to country​ C, and 16 have been only to country A. Some club members have not been to any of the three foreign​ countries, and,​ curiously, an equal number have been to all three countries.
​(a) How many students have been to all three​ countries?

To solve this problem, we'll use the principle of inclusion-exclusion, which is helpful when dealing with overlapping sets. Let’s define the variables and sets:

- Total number of students: \( |U| = 81 \)
- Number of students who visited:
  - Country A: \( |A| = 34 \)
  - Country B: \( |B| = 38 \)
  - Country C: \( |C| = 33 \)
  - Both A and B: \( |A \cap B| = 12 \)
  - Only C: \( |C \setminus (A \cup B)| = 14 \)
  - Only A: \( |A \setminus (B \cup C)| = 16 \)
- Let \( x \) represent the number of students who have visited all three countries.

### Step-by-Step Solution:

1. **Number of students visiting only A**: 
   This is given directly as \( 16 \). 

2. **Number of students visiting only C**: 
   This is also given directly as \( 14 \).

3. **Number of students visiting only B**:
   We need to calculate this indirectly. Let’s start by calculating the number of students who have visited just A and B (excluding C) from \( |A \cap B| = 12 \).

4. **Inclusion-Exclusion Formula**:
   The number of students who visited all three countries is represented by \( x \), and the number of students who have visited any of the three countries is:
   \[
   |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|
   \]
   We'll substitute the known values into the inclusion-exclusion equation. To compute how many visited all three countries, we set:



There are 81 students in the Travel Club. They discovered that 34 members have visited country A, 38 have visited country B, 33 have been to country C, 12 have visited countries A and B, 14 have been only to country C, and 16 have been only to country A. Some club members have not been to any of the three foreign countries, and, curiously, an equal number have been to all three countries. (a) How many students have been to all three countries?

Solve a set theory problem involving 81 students in a travel club and three overlapping sets for countries A, B, and C. Learn how to use the Inclusion-Exclusion Principle to determine how many students visited all three countries. This problem is ideal for students in Grades 10-12.

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