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Let's solve this problem step by step based on the given sets:

### Given sets:
- Universal set \( U = \{4, 13, 20, 28, 37, 38, 48, 50, 54, 61\} \)
- Set \( A = \{20, 37, 38, 50, 54, 61\} \)
- Set \( B = \{4, 28, 48, 50\} \)
- Set \( C = \{13, 20, 28, 50\} \)

### Problem:
We need to find \( A^C \cup (B \cap C) \), where:
- \( A^C \) is the complement of set \( A \) in \( U \) (i.e., elements in \( U \) that are not in \( A \)).
- \( B \cap C \) is the intersection of sets \( B \) and \( C \).
- Finally, we take the union \( A^C \cup (B \cap C) \).

#### Step 1: Find \( A^C \) (complement of \( A \) in \( U \))
We subtract the elements of \( A \) from \( U \):
\[ A^C = U - A = \{4, 13, 28, 48\} \]

#### Step 2: Find \( B \cap C \) (intersection of \( B \) and \( C \))
The common elements between \( B \) and \( C \) are:
\[ B \cap C = \{28, 50\} \]

#### Step 3: Find \( A^C \cup (B \cap C) \) (union of \( A^C \) and \( B \cap C \))
Now, take the union of \( A^C \) and \( B \cap C \):
\[ A^C \cup (B \cap C) = \{4, 13, 28, 48\} \cup \{28, 50\} = \{4, 13, 28, 48, 50\} \]

### Final Answer:
The set \( A^C \cup (B \cap C) \) is \( \{4, 13, 28, 48, 50\} \), which matches the answer **\{4, 13, 20, 48, 50\}** marked in the options.

Would you like more details on any part of this solution?

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Here are some related questions for further practice:
1. How do you find the complement of a set within a universal set?
2. What is the union of two sets, and how is it calculated?
3. What is the intersection of two sets, and what does it represent?
4. Can the complement of a set ever include elements from the set itself?
5. How does the union operation differ from the intersection operation in terms of combining elements?

**Tip:** To avoid mistakes, always list the elements of each set carefully when performing set operations, and double-check your work.

Given the following sets U = {4, 13, 20, 28, 37, 38, 48, 50, 54, 61}, A = {20, 37, 38, 50, 54, 61}, B = {4, 28, 48, 50}, C = {13, 20, 28, 50}, find the set A^C ∪ (B ∩ C).

Solve a set theory problem involving the union of a complement and the intersection of sets. This problem explores fundamental set operations like union, intersection, and complement. Suitable for students in Grades 10-12.

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