1. **Problem Statement:** We have 100 students selling fruits: 40 sell apples, 46 sell oranges, 50 sell mangoes. 14 sell both apples and oranges, 15 sell both apples and mangoes, and 10 sell all three fruits. Every student sells at least one fruit. We want to find the number of students selling only apples, only oranges, only mangoes, and those selling exactly two fruits. 2. **Formula and Rules:** Use the principle of inclusion-exclusion for three sets $A$, $O$, and $M$ representing apples, oranges, and mangoes sellers respectively. $$|A \cup O \cup M| = |A| + |O| + |M| - |A \cap O| - |A \cap M| - |O \cap M| + |A \cap O \cap M|$$ Given $|A \cup O \cup M| = 100$ (all students), and the values for $|A|$, $|O|$, $|M|$, $|A \cap O|$, $|A \cap M|$, and $|A \cap O \cap M|$, we can find $|O \cap M|$ and then the numbers selling only one or exactly two fruits. 3. **Calculate $|O \cap M|$:** $$100 = 40 + 46 + 50 - 14 - 15 - |O \cap M| + 10$$ Simplify: $$100 = 136 - 29 - |O \cap M| + 10$$ $$100 = 117 - |O \cap M|$$ $$|O \cap M| = 117 - 100 = 17$$ 4. **Find numbers selling exactly two fruits:** - Exactly apples and oranges only: $$|A \cap O| - |A \cap O \cap M| = 14 - 10 = 4$$ - Exactly apples and mangoes only: $$|A \cap M| - |A \cap O \cap M| = 15 - 10 = 5$$ - Exactly oranges and mangoes only: $$|O \cap M| - |A \cap O \cap M| = 17 - 10 = 7$$ 5. **Find numbers selling only one fruit:** - Only apples: $$|A| - (\text{exactly } A \cap O + \text{exactly } A \cap M + |A \cap O \cap M|) = 40 - (4 + 5 + 10) = 21$$ - Only oranges: $$|O| - (\text{exactly } A \cap O + \text{exactly } O \cap M + |A \cap O \cap M|) = 46 - (4 + 7 + 10) = 25$$ - Only mangoes: $$|M| - (\text{exactly } A \cap M + \text{exactly } O \cap M + |A \cap O \cap M|) = 50 - (5 + 7 + 10) = 28$$ 6. **Summary:** - Only apples: 21 - Only oranges: 25 - Only mangoes: 28 - Exactly apples and oranges: 4 - Exactly apples and mangoes: 5 - Exactly oranges and mangoes: 7 - All three fruits: 10 These numbers add up to 100, confirming the solution. **Final answer:** Only apples: 21 Only oranges: 25 Only mangoes: 28 Exactly two fruits: apples & oranges = 4, apples & mangoes = 5, oranges & mangoes = 7 All three fruits: 10