**Problem 1:** Simplify the expression $3\sqrt{50} + 2\sqrt{18} - \sqrt{32}$. 1. Start by simplifying each square root. $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ 2. Substitute back into the expression: $3\sqrt{50} + 2\sqrt{18} - \sqrt{32} = 3(5\sqrt{2}) + 2(3\sqrt{2}) - 4\sqrt{2} = 15\sqrt{2} + 6\sqrt{2} - 4\sqrt{2}$ 3. Combine like terms: $15\sqrt{2} + 6\sqrt{2} - 4\sqrt{2} = (15 + 6 - 4)\sqrt{2} = 17\sqrt{2}$ **Answer to Problem 1:** $17\sqrt{2}$ --- **Problem 2:** Use a Venn diagram to represent the medal distribution: 37 countries won gold (G), 44 silver (S), 54 bronze (B), 30 won both gold and silver, 33 won gold and silver (likely a typo here, assuming it's gold and bronze), 36 won silver and bronze, and 28 won all three medals. 1. Let the sets be $G, S, B$. The intersection of all three is $|G \cap S \cap B| = 28$. 2. Use given pairwise intersection data: $|G \cap S| = 30$, $|G \cap B| = 33$, $|S \cap B| = 36$. 3. Calculate those who won exactly two medals (excluding those who won all three): $|G \cap S| - |G \cap S \cap B| = 30 - 28 = 2$ $|G \cap B| - |G \cap S \cap B| = 33 - 28 = 5$ $|S \cap B| - |G \cap S \cap B| = 36 - 28 = 8$ 4. Calculate those who won only one type of medal: $|G| - (|G \cap S| + |G \cap B| - |G \cap S \cap B|) = 37 - (30 + 33 - 28) = 37 - 35 = 2$ $|S| - (|G \cap S| + |S \cap B| - |G \cap S \cap B|) = 44 - (30 + 36 - 28) = 44 - 38 = 6$ $|B| - (|G \cap B| + |S \cap B| - |G \cap S \cap B|) = 54 - (33 + 36 - 28) = 54 - 41 = 13$ **Answer to Problem 2 (i):** The Venn diagram includes these cardinalities: - Only Gold: 2 - Only Silver: 6 - Only Bronze: 13 - Gold & Silver only: 2 - Gold & Bronze only: 5 - Silver & Bronze only: 8 - All three medals: 28 --- **Problem 2 (ii):** How many countries won silver medals only? From the above, the number is $6$. --- **Problem 2 (iii):** How many countries won silver and gold but no bronze? That is $|G \cap S| - |G \cap S \cap B| = 2$. **Final answers:** 1. $17\sqrt{2}$ 2. Venn diagram sets as stated above 3. Silver only: 6 4. Silver and gold but no bronze: 2