1. **State the problem:** We are given a Venn diagram with three sets A, B, and C, and we need to find the cardinality of the set $(A \cap B) \cup C$. 2. **Recall the formula:** The union of two sets $X$ and $Y$ is given by $|X \cup Y| = |X| + |Y| - |X \cap Y|$. 3. **Identify the regions:** From the Venn diagram: - $A \cap B$ includes the numbers in the intersection of A and B only (3) and the intersection of all three sets (5). - Set $C$ includes the numbers in C only (8), the intersection of A and C only (2), the intersection of B and C only (7), and the intersection of all three sets (5). 4. **Calculate $|A \cap B|$:** $$|A \cap B| = 3 + 5 = 8$$ 5. **Calculate $|C|$:** $$|C| = 8 + 2 + 7 + 5 = 22$$ 6. **Calculate $|(A \cap B) \cap C|$:** This is the intersection of all three sets, which is 5. 7. **Apply the union formula:** $$|(A \cap B) \cup C| = |A \cap B| + |C| - |(A \cap B) \cap C|$$ $$= 8 + 22 - 5$$ 8. **Simplify:** $$= 30 - 5 = 25$$ **Final answer:** $$\boxed{25}$$