We are given the following information about the sets:

• $n(A) = 26$
• $n(B) = 30$
• $n(C) = 23$
• $n(A \cap B) = 10$
• $n(B \cap C) = 11$
• $n(A \cap C) = 7$
• $n(A \cap B \cap C) = 4$
• $n(U) = 64$, where $U$ is the universal set.

The Venn diagram regions are labeled, and the goal is to fill them with appropriate numbers. Let's break this down by solving for each region:

Overlapping Regions:

• Region V (A ∩ B ∩ C): This is the overlap of all three sets. We are given that $n(A \cap B \cap C) = 4$, so this region contains 4 elements.

• Region VI (A ∩ B, but not C): This is the part of A and B that does not include C. We know $n(A \cap B) = 10$, and 4 of these are in $A \cap B \cap C$. Therefore, $10 - 4 = 6$ elements are in region VI.

• Region VII (A ∩ C, but not B): Similarly, $n(A \cap C) = 7$, and 4 of these are in $A \cap B \cap C$, so $7 - 4 = 3$ elements are in region VII.

• Region VIII (B ∩ C, but not A): $n(B \cap C) = 11$, and 4 are in $A \cap B \cap C$, so $11 - 4 = 7$ elements are in region VIII.

Non-overlapping Regions:

• Region II (Only A): We know that $n(A) = 26$, and we’ve already accounted for the elements in $A \cap B$, $A \cap C$, and $A \cap B \cap C$. The total in these overlaps is $6 + 3 + 4 = 13$. Therefore, the number of elements in region II (only A) is $26 - 13 = 13$.

• Region IV (Only B): $n(B) = 30$, and the overlaps involving B (regions VI, V, and VIII) have $6 + 4 + 7 = 17$ elements. So, region IV (only B) contains $30 - 17 = 13$ elements.

• Region III (Only C): $n(C) = 23$, and the overlaps involving C (regions VII, V, and VIII) have $3 + 4 + 7 = 14$ elements. Thus, region III (only C) contains $23 - 14 = 9$ elements.

Outside the Sets (Region I):

The total number of elements in the universal set is $n(U) = 64$. We have accounted for all the elements inside A, B, and C, which are:

$13 \, (\text{only A}) + 13 \, (\text{only B}) + 9 \, (\text{only C}) + 6 \, (A \cap B) + 3 \, (A \cap C) + 7 \, (B \cap C) + 4 \, (A \cap B \cap C) = 55$

Thus, the number of elements in region I (outside A, B, and C) is $64 - 55 = 9$.

Summary:

• Region I: 9 elements
• Region II: 13 elements
• Region III: 9 elements
• Region IV: 13 elements
• Region V: 4 elements
• Region VI: 6 elements
• Region VII: 3 elements
• Region VIII: 7 elements

Would you like further details or clarification?

Here are some related questions:

1. What is the total number of elements shared between exactly two sets?
2. How would the diagram change if $n(U)$ were increased?
3. What happens if one of the sets has no overlap with the others?
4. How do you compute the number of elements in only two sets?
5. What if $n(A \cap B \cap C)$ was 0—how would the regions be affected?

Tip: When working with Venn diagrams, always start with the intersection of all three sets and work outward to ensure no overlap is missed.