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: