To solve the problem of finding $n[ A \cup (B \cap C)^c ]$ using the given Venn diagram, we will break it down step by step:

1. Interpretation of the problem:

   • $A \cup (B \cap C)^c$ means the union of set $A$ with the complement of the intersection of sets $B$ and $C$.
   • $(B \cap C)^c$ is everything outside the intersection of sets $B$ and $C$.

2. Step-by-step approach:

   • First, identify $B \cap C$ from the diagram:
     • The intersection of $B$ and $C$ is the area that includes 24, 17, and 12 (shared between $B$ and $C$).
     • So, $B \cap C = \{12, 24, 17\}$.
   • Next, find $(B \cap C)^c$, which includes:
     • Everything except the elements 12, 24, and 17. Therefore, $(B \cap C)^c = \{10, 13, 3, 10, 21\}$.
   • Now, find $A \cup (B \cap C)^c$, which is:
     • The union of set $A$ and $(B \cap C)^c$.
     • Set $A$ includes: $\{10, 13, 12, 24\}$ (from the diagram).
     • The union of $A$ with $(B \cap C)^c$ gives $\{10, 13, 12, 24, 3, 21\}$.

3. Counting the elements:

   • Unique elements in $A \cup (B \cap C)^c$ are $\{10, 13, 12, 24, 3, 21\}$.
   • The total number of unique elements is 6.

4. Total count of elements in the universal set:

   • Add all the values in the universal set: $10 + 13 + 12 + 24 + 10 + 17 + 3 + 21 = 110$.
   • Now subtract the elements of $B \cap C$, which are: $12 + 24 + 17 = 53$.
   • Subtract these from the total: $110 - 53 = 57$.

Thus, the answer is none of the provided options.