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We are tasked with determining the number of elements in each region of the Venn diagram. The cardinalities provided are:

- \( n(U) = 44 \)
- \( n(A) = 18 \)
- \( n(B) = 22 \)
- \( n(C) = 18 \)
- \( n(A \cap B) = 11 \)
- \( n(A \cap C) = 8 \)
- \( n(B \cap C) = 7 \)
- \( n(A \cap B \cap C) = 3 \) (already labeled in region V)

### Step-by-Step Process:

We will calculate the number of elements in each region, using the Venn diagram labels:

#### Region V (already given as 3):
- Region V represents \( A \cap B \cap C \), which is already provided as 3.

#### Region II (inside \( A \cap B \) but not \( C \)):
- \( n(A \cap B) = 11 \), which includes Region V. Thus, Region II = \( 11 - 3 = 8 \).

#### Region IV (inside \( A \cap C \) but not \( B \)):
- \( n(A \cap C) = 8 \), which also includes Region V. Thus, Region IV = \( 8 - 3 = 5 \).

#### Region VI (inside \( B \cap C \) but not \( A \)):
- \( n(B \cap C) = 7 \), which also includes Region V. Thus, Region VI = \( 7 - 3 = 4 \).

#### Region I (inside \( A \) but not \( B \) or \( C \)):
- \( n(A) = 18 \), which includes Regions II, IV, and V. Thus, Region I = \( 18 - (8 + 5 + 3) = 2 \).

#### Region III (inside \( B \) but not \( A \) or \( C \)):
- \( n(B) = 22 \), which includes Regions II, VI, and V. Thus, Region III = \( 22 - (8 + 4 + 3) = 7 \).

#### Region VII (inside \( C \) but not \( A \) or \( B \)):
- \( n(C) = 18 \), which includes Regions IV, VI, and V. Thus, Region VII = \( 18 - (5 + 4 + 3) = 6 \).

#### Region VIII (outside all sets):
- \( n(U) = 44 \) represents the total number of elements. To find Region VIII, we subtract the sum of all other regions:
  
  \( 44 - (2 + 8 + 7 + 5 + 3 + 4 + 6) = 44 - 35 = 9 \).

### Final Answers:
- Region I = 2
- Region II = 8
- Region III = 7
- Region IV = 5
- Region VI = 4
- Region VII = 6
- Region VIII = 9

Would you like a detailed breakdown of these regions or further explanation?

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Here are 5 follow-up questions to expand your understanding:
1. How do we calculate the number of elements in the union of sets \( A \), \( B \), and \( C \)?
2. What happens if two sets are disjoint? How would that change the diagram?
3. How can we verify that our solution is correct using the principle of inclusion-exclusion?
4. What is the significance of Region VIII in the context of the universal set \( U \)?
5. Can you think of a real-world example where Venn diagrams might be used to represent overlapping groups?

**Tip:** Always start by calculating the intersection regions first in Venn diagram problems, as they help clarify the more straightforward regions.

The accompanying Venn diagram shows the number of elements in region V. Use the given cardinalities to determine the number of elements in each of the other seven regions.

Learn how to solve a Venn diagram problem involving three sets A, B, and C. Using given cardinalities, we calculate the number of elements in each region using the Inclusion-Exclusion Principle. This problem is suitable for high school students studying set theory and Venn diagrams.

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