Let’s solve this problem step by step.

Given:

• $n(P) = 5$
• $n(Q) = 9$
• $n(P \cup Q) = 10$

We need to find $n(P \cap Q)$, the number of elements in the intersection of $P$ and $Q$.

Formula for Union of Two Sets:

$n(P \cup Q) = n(P) + n(Q) - n(P \cap Q)$

Substitute the given values: $10 = 5 + 9 - n(P \cap Q)$

Simplify: $10 = 14 - n(P \cap Q)$

Rearrange to solve for $n(P \cap Q)$: $n(P \cap Q) = 14 - 10 = 4$

Final Answer:

$\boxed{4}$

Would you like further clarification or additional examples? Here are five related questions to expand your understanding:

1. How do you calculate the union of three sets?
2. If $n(P \cap Q) = 0$, what does this imply about sets $P$ and $Q$?
3. How can you represent Venn diagrams for more than two sets mathematically?
4. What changes in the formula if $P$ and $Q$ are subsets of a universal set $U$?
5. Can the value of $n(P \cup Q)$ ever be smaller than $n(P)$ or $n(Q)$?

Tip: Always verify your answers using the basic set theory formula for union and intersection to avoid calculation errors!