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Let's break down the information and solve the problem.

We have a Venn diagram involving three sets \( A \), \( B \), and \( C \) along with their cardinalities (number of elements). The following data is provided:

- \( n(A) = 26 \)
- \( n(B) = 30 \)
- \( n(C) = 23 \)
- \( n(A \cap B) = 10 \)
- \( n(B \cap C) = 11 \)
- \( n(A \cap C) = 7 \)
- \( n(A \cap B \cap C) = 4 \)
- \( n(U) = 64 \) (The total number of elements in the universal set \( U \))

### Labeling the regions
The Venn diagram regions are labeled as follows:
- I: Outside of all sets
- II: Only in \( A \)
- III: Only in \( C \)
- IV: Only in \( B \)
- V: In all three sets (\( A \cap B \cap C \))
- VI: In \( A \) and \( B \), but not \( C \)
- VII: In \( A \) and \( C \), but not \( B \)
- VIII: In \( B \) and \( C \), but not \( A \)

### Using the data to fill in the regions:
- \( n(A \cap B \cap C) = 4 \) corresponds to region V.
- \( n(A \cap B) = 10 \), so region VI has \( 10 - 4 = 6 \) elements (since 4 elements are already in region V).
- \( n(B \cap C) = 11 \), so region VIII has \( 11 - 4 = 7 \) elements.
- \( n(A \cap C) = 7 \), so region VII has \( 7 - 4 = 3 \) elements.

Now we calculate the remaining elements in sets \( A \), \( B \), and \( C \):
- \( n(A) = 26 \), so region II has \( 26 - (6 + 4 + 3) = 13 \) elements.
- \( n(B) = 30 \), so region IV has \( 30 - (6 + 4 + 7) = 13 \) elements.
- \( n(C) = 23 \), so region III has \( 23 - (3 + 4 + 7) = 9 \) elements.

Finally, we calculate the number of elements in region I (outside all sets):
- \( n(U) = 64 \), so region I has \( 64 - (13 + 6 + 3 + 13 + 9 + 7 + 4) = 9 \) elements.

### Solution for Region II:
- Region II contains 13 elements.

Would you like a breakdown of any specific parts, or do you have additional questions?

Here are some related questions for practice:
1. What is the number of elements in regions VII and VIII combined?
2. How many elements are present in \( B \cup C \)?
3. What is the number of elements in the set \( (A \cup B \cup C)^c \) (outside all sets)?
4. How would the numbers change if \( n(A \cap B \cap C) \) increased to 5?
5. How many elements are only in \( A \cup C \), but not in \( B \)?

**Tip:** In Venn diagrams with three sets, breaking the diagram into distinct regions based on the given intersections simplifies the process of filling in values.

Fill in the Venn diagram with the appropriate numbers based on the following information:

n(A) = 26
n(B) = 30
n(C) = 23
n(A ∩ B) = 10
n(B ∩ C) = 11
n(A ∩ C) = 7
n(A ∩ B ∩ C) = 4
n(U) = 64

Determine how many elements are in each region of the Venn diagram, and calculate the number of elements in Region II.

This problem focuses on calculating the number of elements in a Venn diagram with three sets, A, B, and C. Using set theory concepts and the Principle of Inclusion-Exclusion, learn how to solve for individual regions in the diagram. Suitable for Grades 8-10.

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