1. **Problem Statement:** We have a survey of 50 people about their lunch preferences among three hotels: Hilltop, Serena, and Lemigo. Given are the total numbers who ate at each hotel and their intersections. 2. **Given Data:** - Total people surveyed: 50 - Hilltop (H): 15 - Serena (S): 30 - Lemigo (L): 19 - Hilltop and Serena (H \cap S): 8 - Hilltop and Lemigo (H \cap L): 12 - Serena and Lemigo (S \cap L): 7 - All three (H \cap S \cap L): 5 3. **Step (a): Represent the information on a Venn diagram** - The Venn diagram has three intersecting circles labeled H, S, and L. - The intersection of all three is 5. - To find the number of people in exactly two sets, subtract those in all three from the pairwise intersections: $$|H \cap S| - |H \cap S \cap L| = 8 - 5 = 3$$ $$|H \cap L| - |H \cap S \cap L| = 12 - 5 = 7$$ $$|S \cap L| - |H \cap S \cap L| = 7 - 5 = 2$$ - To find those who ate at only one hotel, subtract the intersections from the totals: $$|H| - (|H \cap S| + |H \cap L| - |H \cap S \cap L|) = 15 - (8 + 12 - 5) = 15 - 15 = 0$$ $$|S| - (|H \cap S| + |S \cap L| - |H \cap S \cap L|) = 30 - (8 + 7 - 5) = 30 - 10 = 20$$ $$|L| - (|H \cap L| + |S \cap L| - |H \cap S \cap L|) = 19 - (12 + 7 - 5) = 19 - 14 = 5$$ 4. **Step (b): How many people ate at Hilltop only?** From above, Hilltop only is 0. 5. **Step (c): How many people ate at Hilltop and Serena but not at Lemigo?** This is the number in the intersection of Hilltop and Serena excluding Lemigo: $$|H \cap S| - |H \cap S \cap L| = 8 - 5 = 3$$ **Final answers:** - Hilltop only: 0 - Hilltop and Serena but not Lemigo: 3 Note: The numbers inside the Venn diagram regions are: - Hilltop only: 0 - Serena only: 20 - Lemigo only: 5 - Hilltop and Serena only: 3 - Hilltop and Lemigo only: 7 - Serena and Lemigo only: 2 - All three: 5