Given:

• $n(U) = 200$ (the total number of elements in the universal set $U$),
• $n(A \cup B) = 150$ (the number of elements in the union of sets $A$ and $B$).

We are asked to find $n(A^c \cap B^c)$, which represents the number of elements in the intersection of the complements of sets $A$ and $B$. This intersection is also the complement of $A \cup B$, denoted as $(A \cup B)^c$.

Using the principle of complements:

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

Substituting the given values:

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

So, $n(A^c \cap B^c) = 50$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the principle of inclusion-exclusion?
2. How do you calculate $n(A \cap B)$ given $n(A)$, $n(B)$, and $n(A \cup B)$?
3. What is the significance of complements in set theory?
4. Can you explain the difference between $A \cup B$ and $A \cap B$?
5. How do Venn diagrams help in understanding set operations?

Tip: When working with complements, remember that the complement of a union is the intersection of the complements, and vice versa.