n(A) = 29, n(B) = 37, n(C) = 37, n(A ∩ B) = 17, n(U) = 77

The provided symbols indicate the cardinalities (sizes) of sets \(A\), \(B\), \(C\), their intersections, and the universal set \(U\). However, there is missing information about \(C\)'s interaction with other sets. Let's address what we can solve with the provided information:

### Given:
- \(n(A) = 29\) (Number of elements in set \(A\))
- \(n(B) = 37\) (Number of elements in set \(B\))
- \(n(C) = 37\) (Number of elements in set \(C\))
- \(n(A \cap B) = 17\) (Number of elements in both sets \(A\) and \(B\))
- \(n(U) = 77\) (Total number of elements in the universal set \(U\))

### Let's compute \(n(A \cup B)\):
Using the inclusion-exclusion principle:
\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]
Substitute the values:
\[
n(A \cup B) = 29 + 37 - 17 = 49
\]
This means the number of elements in the union of sets \(A\) and \(B\) is 49.

### What's still unknown:
1. The interaction of set \(C\) with sets \(A\) and \(B\) (i.e., intersections \(A \cap C\), \(B \cap C\), or \(A \cap B \cap C\)) is not given. Without this, we can't fully analyze the relationships between all three sets.

### Questions:
1. Do you have information about intersections involving set \(C\)?
2. Do sets \(A\) and \(B\) have any interaction with set \(C\)?
3. Would you like to calculate specific regions of a Venn diagram for these sets?
4. Do you need to find how many elements belong exclusively to each set?
5. Would you like to visualize the solution using a Venn diagram?

**Tip:** When working with multiple sets, the inclusion-exclusion principle helps account for overlapping elements to avoid double-counting.

Solve a set theory problem involving the cardinalities of sets A, B, and their intersection, using the inclusion-exclusion principle. Learn how to calculate n(A ∪ B) with given values n(A) = 29, n(B) = 37, n(A ∩ B) = 17, and n(U) = 77. Suitable for high school students (Grades 9-12).

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