1. **Problem Statement:** Happy Elementary has three clubs: Art (A), Book (B), and Computer (C) with given membership counts and intersections. We need to analyze membership counts, draw a Venn diagram, and answer questions about membership. 2. **Given Data:** - Total children, $|U| = 185$ - $|A| = 66$ - $|B| = 120$ - $|C| = 78$ - $|A \cap B \cap C| = 20$ - $|A \cap B| = 60$ - $|B \cap C| = 40$ - $|A \cap C| = 23$ 3. **Step 1: Find the number of children in exactly two clubs.** We know the triple intersection is included in each pair intersection, so: $$|A \cap B|_{exactly} = |A \cap B| - |A \cap B \cap C| = 60 - 20 = 40$$ $$|B \cap C|_{exactly} = 40 - 20 = 20$$ $$|A \cap C|_{exactly} = 23 - 20 = 3$$ 4. **Step 2: Find the number of children in exactly one club.** Use the formula: $$|A| = |A_{only}| + |A \cap B|_{exactly} + |A \cap C|_{exactly} + |A \cap B \cap C|$$ So, $$|A_{only}| = |A| - (|A \cap B|_{exactly} + |A \cap C|_{exactly} + |A \cap B \cap C|) = 66 - (40 + 3 + 20) = 3$$ Similarly for $B$: $$|B_{only}| = 120 - (40 + 20 + 20) = 40$$ For $C$: $$|C_{only}| = 78 - (3 + 20 + 20) = 35$$ 5. **Step 3: Find the number of children not in any club.** Sum all disjoint regions inside the Venn diagram: $$3 + 40 + 35 + 40 + 20 + 3 + 20 = 161$$ Children not in any club: $$|U| - 161 = 185 - 161 = 24$$ 6. **Step 4: Number of children in exactly two clubs:** Sum of exactly two clubs: $$40 + 20 + 3 = 63$$ 7. **Step 5: Describe the shaded region for (d):** The shaded region is children who are not in Computer club or who are in Book club. In set notation: $$\overline{C} \cup B$$ --- **Final answers:** - (b) Children not in any club: $24$ - (c) Children in exactly two clubs: $63$ - (d) Shaded region: $\overline{C} \cup B$ **Venn diagram regions (number of children):** - $A_{only} = 3$ - $B_{only} = 40$ - $C_{only} = 35$ - $A \cap B$ only = 40 - $B \cap C$ only = 20 - $A \cap C$ only = 3 - $A \cap B \cap C = 20$ - Outside all sets = 24