To solve for $n[A \cup (B \cap C)^c]$, we will need to analyze the Venn diagram provided.

Definitions:

1. $A \cup (B \cap C)^c$ is the union of set $A$ and the complement of the intersection of sets $B$ and $C$.
2. The complement of $B \cap C$, written as $(B \cap C)^c$, includes all elements that are not in both $B$ and $C$.

Step-by-step breakdown:

1. Find $B \cap C$:

   • The elements in both $B$ and $C$ are: $11$ and $3$.
   • So, $B \cap C = \{11, 3\}$.

2. Find $(B \cap C)^c$:

   • The complement $(B \cap C)^c$ will be all elements that are not in both $B$ and $C$. This includes:
     • $7$ (only in $A$),
     • $21$ (only in $C$),
     • $13$ (only in $C$),
     • $3$ (only in $A$),
     • $11$ (already counted).
     • The element $19$ from the universal set $U$, which is outside all sets $A$, $B$, and $C$, is also included in $(B \cap C)^c$.

3. Now find $A \cup (B \cap C)^c$:

   • The union of $A$ and $(B \cap C)^c$ means taking all the elements in either set $A$ or $(B \cap C)^c$. From the diagram:
     • $A = \{7, 3, 11\}$.
     • $(B \cap C)^c = \{7, 21, 13, 19\}$ (as found earlier).
   • The union of these sets is: $A \cup (B \cap C)^c = \{7, 3, 11, 21, 13, 19\}$

4. Count the elements in $A \cup (B \cap C)^c$:

   • The set $A \cup (B \cap C)^c$ contains: $\{7, 3, 11, 21, 13, 19\}$, which are 6 distinct elements.

Thus, $n[A \cup (B \cap C)^c] = 6$, which corresponds to None of the above from the given choices.

Would you like more details or further clarifications?

Here are 5 relative questions for deeper understanding:

1. What is the role of the universal set $U$ in this problem?
2. How do we define set complements in the context of Venn diagrams?
3. How would the result change if the universal set contained different elements?
4. Can you explain the difference between union and intersection in set theory?
5. How can we apply this concept to real-life situations like survey data?

Tip: Always break down set operations like complements and unions step by step to avoid confusion, especially when dealing with intersections and their complements.