We are given the following information from the question:

• $n(U) = 60$ (the total number of elements in the universal set),
• $n(A) = 35$ (the number of elements in set A),
• $n(B) = 24$ (the number of elements in set B),
• $n(A \cup B)' = 10$ (the number of elements outside of the union of A and B).

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

Step 1: Calculate $n(A \cup B)$

From the information given, $n(A \cup B)' = 10$, meaning the number of elements outside of $A \cup B$ is 10. Since the universal set $U$ contains 60 elements, we can find the number of elements in $A \cup B$ by subtracting $n(A \cup B)'$ from $n(U)$.

$n(A \cup B) = n(U) - n(A \cup B)' = 60 - 10 = 50$

Step 2: Verify if the answer is one of the options

The number of elements in $A \cup B$ is $50$. None of the provided options (9, 8, 6) match this value. Therefore, the correct answer is indeed "None of these."

Would you like more details, or do you have any questions?

Here are five related questions you might want to explore:

1. What is the formula for the union of two sets in terms of their individual elements and intersections?
2. How would you calculate the number of elements in $A \cup B$ if the number of elements in $A \cap B$ is given?
3. What is the complement of a set, and how is it used in set theory problems like this?
4. How does the inclusion-exclusion principle apply to the union of two sets?
5. How can Venn diagrams help visualize the union and complement of sets?

Tip: Always ensure to subtract complements from the total universal set to find the correct union values when given the complement count.