1. **Problem Statement:** We have a survey of 500 students with percentages liking football (F), hockey (H), and basketball (B). Given data: - $|F|=49\%$ of 500 = 245 students - $|H|=53\%$ of 500 = 265 students - $|B|=62\%$ of 500 = 310 students - $|F \cap H|=27\%$ of 500 = 135 students - $|H \cap B|=29\%$ of 500 = 145 students - $|F \cap B|=28\%$ of 500 = 140 students - $|\text{None}|=5\%$ of 500 = 25 students 2. **Goal:** (a) Find number of students who like all three games $|F \cap H \cap B|$. (b) Find ratio of students who like only football to those who like only hockey. (c) Find number of students who like only one of the three games. (d) Find number of students who like at least two games. 3. **Formula and Rules:** Use the principle of inclusion-exclusion for three sets: $$|F \cup H \cup B| = |F| + |H| + |B| - |F \cap H| - |H \cap B| - |F \cap B| + |F \cap H \cap B|$$ Also, total students = 500, and 5% like none, so: $$|F \cup H \cup B| = 500 - 25 = 475$$ 4. **Find $|F \cap H \cap B|$:** Substitute values: $$475 = 245 + 265 + 310 - 135 - 145 - 140 + |F \cap H \cap B|$$ Calculate sum and differences: $$475 = 820 - 420 + |F \cap H \cap B|$$ $$475 = 400 + |F \cap H \cap B|$$ So, $$|F \cap H \cap B| = 475 - 400 = 75$$ 5. **Find students who like only one game:** Only football: $$|F| - |F \cap H| - |F \cap B| + |F \cap H \cap B| = 245 - 135 - 140 + 75 = 45$$ Only hockey: $$|H| - |F \cap H| - |H \cap B| + |F \cap H \cap B| = 265 - 135 - 145 + 75 = 60$$ Only basketball: $$|B| - |F \cap B| - |H \cap B| + |F \cap H \cap B| = 310 - 140 - 145 + 75 = 100$$ Total only one game: $$45 + 60 + 100 = 205$$ 6. **Find ratio of only football to only hockey:** $$\text{Ratio} = \frac{45}{60} = \frac{3}{4}$$ 7. **Find number of students who like at least two games:** At least two means students in exactly two games plus those in all three. Number in exactly two games: $$|F \cap H| - |F \cap H \cap B| = 135 - 75 = 60$$ $$|H \cap B| - |F \cap H \cap B| = 145 - 75 = 70$$ $$|F \cap B| - |F \cap H \cap B| = 140 - 75 = 65$$ Sum exactly two games: $$60 + 70 + 65 = 195$$ Add those who like all three: $$195 + 75 = 270$$ **Final answers:** (a) 75 students like all three games. (b) Ratio of only football to only hockey is 3:4. (c) 205 students like only one game. (d) 270 students like at least two games.