1. **Problem Statement:** We have three music preferences among students: Jazz, Reggae, and Funk. Given: - $|J|=30$, $|R|=20$, $|F|=20$ - $|J \cap R|=10$, $|J \cap F|=10$, $|R \cap F|=9$ - $|J \cap R \cap F|=3$ - $|\text{None}|=32$ Find: (a) Number who liked Jazz only (b) Number who liked Reggae only (c) Number who liked Funk only (d) Total number of students interviewed 2. **Formula and Rules:** Use the principle of inclusion-exclusion for three sets: $$|J \cup R \cup F| = |J| + |R| + |F| - |J \cap R| - |J \cap F| - |R \cap F| + |J \cap R \cap F|$$ The number who liked only one type is: $$|J \text{ only}| = |J| - |J \cap R| - |J \cap F| + |J \cap R \cap F|$$ (similarly for Reggae and Funk) 3. **Calculate Jazz only:** $$|J \text{ only}| = 30 - 10 - 10 + 3 = 13$$ 4. **Calculate Reggae only:** $$|R \text{ only}| = 20 - 10 - 9 + 3 = 4$$ 5. **Calculate Funk only:** $$|F \text{ only}| = 20 - 10 - 9 + 3 = 4$$ 6. **Calculate total who liked at least one type:** $$|J \cup R \cup F| = 30 + 20 + 20 - 10 - 10 - 9 + 3 = 44$$ 7. **Calculate total number of students interviewed:** $$\text{Total} = |J \cup R \cup F| + |\text{None}| = 44 + 32 = 76$$ **Final answers:** - (a) Jazz only = 13 - (b) Reggae only = 4 - (c) Funk only = 4 - (d) Total students = 76