1. **Problem statement:** In a class of 50 students, 30 like math, 25 like science, and 10 like neither. We need to find: - Number of students who like both subjects. - Number of students who like only math. - Number of students who like only one subject. 2. **Formula and rules:** Let $M$ be the set of students who like math, $S$ be the set who like science, and $N$ be the total number of students. We know: $$|M| = 30, \quad |S| = 25, \quad |N| = 50, \quad |\text{None}| = 10$$ The number of students who like at least one subject is: $$|M \cup S| = N - |\text{None}| = 50 - 10 = 40$$ Using the principle of inclusion-exclusion: $$|M \cup S| = |M| + |S| - |M \cap S|$$ 3. **Find the number who like both subjects:** $$|M \cap S| = |M| + |S| - |M \cup S| = 30 + 25 - 40 = 15$$ 4. **Find the number who like only math:** $$|M \text{ only}| = |M| - |M \cap S| = 30 - 15 = 15$$ 5. **Find the number who like only one subject:** This is the sum of students who like only math and only science. First, find only science: $$|S \text{ only}| = |S| - |M \cap S| = 25 - 15 = 10$$ Then, $$|\text{Only one subject}| = |M \text{ only}| + |S \text{ only}| = 15 + 10 = 25$$ **Final answers:** - Both subjects: 15 - Only math: 15 - Only one subject: 25