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The image shows a Venn diagram problem with three sets, \( A \), \( B \), and \( C \), representing students who have seen different movies. Here is the relevant information extracted from the image:

- \( A \) represents students who have seen Movie A.
- \( B \) represents students who have seen Movie B.
- \( C \) represents students who have seen Movie C.
- The problem provides the following dataThe image shows a Venn diagram problem with three sets, \( A \), \( B \), and \( C \), representing students who have seen different movies. Here is the relevant information extracted from the image:

- \( A \) represents students who have seen Movie A.
- \( B \) represents students who have seen Movie B.
- \( C \) represents students who have seen Movie C.
- The problem provides the following 
  - 43 students had seen Movie A.
  - 29 students had seen Movie B.
  - 29 students had seen Movie C.
  - 16 students had seen both Movies A and B.
  - 13 students had seen both Movies A and C.
  - 11 students had seen both Movies B and C.
  - 4 students had seen all three movies (A, B, and C).
  - 5 students had seen none of the movies.

To solve this problem and complete the Venn diagram, we need to distribute these numbers into the respective regions of the diagram. Let's break it down step by step.

### Step 1: Fill in the intersection of all three sets.
- The number of students who saw all three movies (A, B, and C) is 4, which should be placed in the center of the Venn diagram (intersection of A, B, and C).

### Step 2: Fill in the pairwise intersections.
- \( A \cap B \): 16 students saw both Movies A and B. Since 4 of them already saw all three movies, the number of students who saw only Movies A and B (but not C) is \( 16 - 4 = 12 \).
- \( A \cap C \): 13 students saw both Movies A and C. Again, 4 of them are already counted in all three, so \( 13 - 4 = 9 \) students saw only Movies A and C (but not B).
- \( B \cap C \): 11 students saw both Movies B and C. Subtracting the 4 who saw all three movies, \( 11 - 4 = 7 \) students saw only Movies B and C (but not A).

### Step 3: Fill in the remaining parts of each set.
- For Movie A:
  - 43 students saw Movie A in total.
  - We already accounted for \( 12 + 9 + 4 = 25 \) students in the intersections.
  - Therefore, the number of students who saw only Movie A is \( 43 - 25 = 18 \).
- For Movie B:
  - 29 students saw Movie B in total.
  - We already accounted for \( 12 + 7 + 4 = 23 \) students in the intersections.
  - So, the number of students who saw only Movie B is \( 29 - 23 = 6 \).
- For Movie C:
  - 29 students saw Movie C in total.
  - We accounted for \( 9 + 7 + 4 = 20 \) students in the intersections.
  - So, the number of students who saw only Movie C is \( 29 - 20 = 9 \).

### Step 4: Fill in the number of students outside the sets.
- 5 students saw none of the movies, so this goes outside the three circles of the Venn diagram.

### Final Distribution:
- Only A: 18 students.
- Only B: 6 students.
- Only C: 9 students.
- \( A \cap B \) (only): 12 students.
- \( A \cap C \) (only): 9 students.
- \( B \cap C \) (only): 7 students.
- \( A \cap B \cap C \): 4 students.
- Outside all sets: 5 students.

Would you like more details or have any questions?

### Related Questions:
1. How do we calculate the total number of students in the group?
2. Can we find the probability of a student having seen at least one movie?
3. What is the probability that a randomly selected student saw exactly two movies?
4. How many students saw exactly one movie?
5. How do Venn diagrams help in solving set theory problems like this?

#### Tip:
When working with Venn diagrams, always start by filling in the intersections and work your way outwards for an organized approach.

A middle school counselor, attempting to correlate school performance with leisure interests, found that of a group of students, 43 had seen Movie A, 29 had seen Movie B, and 29 had seen Movie C. 16 had seen Movies A and B, 13 had seen Movies A and C, 11 had seen Movies B and C, 4 had seen all three films, and 5 had seen none of the three films. Use a Venn diagram to complete parts (a) through (c).

Solve a Venn diagram problem involving students watching three movies (A, B, and C). Learn how to distribute 43 students who saw Movie A, 29 students for Movies B and C, and handle intersections using the inclusion-exclusion principle. The problem also covers basic set theory and probability.

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