1. **Problem statement:** In a class of 50 students, 30 like mango, 25 like guava, and 10 like none of the fruits. Find the number of students who like both fruits, mango only, and only one of the fruits. 2. **Formula and rules:** - Let $M$ be the set of students who like mango, $G$ be the set who like guava. - Total students $= 50$. - Number who like none $= 10$. - Number who like either or both fruits $= 50 - 10 = 40$. - Using the principle of inclusion-exclusion: $$|M \cup G| = |M| + |G| - |M \cap G|$$ 3. **Find number who like both fruits:** $$40 = 30 + 25 - |M \cap G|$$ $$|M \cap G| = 30 + 25 - 40 = 15$$ 4. **Find number who like mango only:** $$|M \text{ only}| = |M| - |M \cap G| = 30 - 15 = 15$$ 5. **Find number who like only one fruit:** $$|M \text{ only}| + |G \text{ only}| = |M| - |M \cap G| + |G| - |M \cap G| = (30 - 15) + (25 - 15) = 15 + 10 = 25$$ **Final answers:** - Number who like both fruits $= 15$ - Number who like mango only $= 15$ - Number who like only one fruit $= 25$