Given:

• $n(A \cap B) = 50$
• $n(A \cap B^c) = 200$
• $n(A^c \cap B) = 150$

We need to find $n(A \cup B)$, the number of elements in the union of sets $A$ and $B$.

Steps to find $n(A \cup B)$:

1. Break down the union into disjoint parts:

   The union of $A$ and $B$ can be expressed as the sum of the following disjoint sets:

   • $A \cap B$ (elements in both $A$ and $B$)
   • $A \cap B^c$ (elements in $A$ but not in $B$)
   • $A^c \cap B$ (elements in $B$ but not in $A$)

   Thus:

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

2. Substitute the given values:

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

3. Calculate the total:

   $n(A \cup B) = 400$

So, $n(A \cup B) = 400$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find $n(A \cup B)$ using the principle of inclusion-exclusion?
2. What is the significance of the disjoint parts in set operations?
3. How would the calculation change if you had $n(U)$ (total number of elements in the universal set)?
4. Can you explain how to visualize this problem using a Venn diagram?
5. What are some practical applications of set operations like union and intersection?

Tip: When dealing with unions and intersections, breaking down the sets into disjoint parts can simplify the calculation.