If n(A∩B) = 50, n(A∩Bc) = 200, n(Ac∩B) = 150, what is n(A∪B)?

To find \( n(A \cup B) \), which represents the number of elements in the union of sets \( A \) and \( B \), we will use the information provided:

- \( n(A \cap B) = 50 \): Number of elements in both \( A \) and \( B \).
- \( n(A \cap B^c) = 200 \): Number of elements in \( A \) but not in \( B \) (here, \( B^c \) represents the complement of \( B \)).
- \( n(A^c \cap B) = 150 \): Number of elements in \( B \) but not in \( A \) (here, \( A^c \) represents the complement of \( A \)).

### Step-by-Step Solution

The total number of elements in the union of \( A \) and \( B \) is given by:

\[
n(A \cup B) = n(A \cap B) + n(A \cap B^c) + n(A^c \cap B)
\]

Substitute the given values:

\[
n(A \cup B) = 50 + 200 + 150
\]

Calculate the sum:

\[
n(A \cup B) = 400
\]

### Conclusion

The number of elements in the union of sets \( A \) and \( B \) is:

\[
\boxed{400}
\]

Would you like more details or have any questions?

### Related Questions
1. How would you calculate \( n(A \cup B) \) if you were given \( n(A) \) and \( n(B) \) instead?
2. How do you determine \( n(A^c \cup B^c) \) using the information given?
3. What is the relationship between \( n(A \cup B) \) and \( n(A^c \cap B^c) \)?
4. How would \( n(A \cup B) \) change if \( n(A \cap B^c) \) were 300 instead of 200?
5. What formula would you use if you wanted to find \( n(A \cap B^c) + n(A^c \cap B^c) \)?

**Tip:** When working with set operations, always consider using Venn diagrams to visualize the different regions, which can help make complex problems more manageable.

Explore how to calculate the union of sets A and B using given intersection values. Understand set theory concepts with step-by-step solution and formulas. Suitable for Grades 10-12.

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