1. **State the problem:** We have 120 students surveyed about their study of Mathematics (M), Economics (E), and Computer Studies (C). Given the numbers studying each subject and their overlaps, we want to analyze the data. 2. **Given data:** - $|M|=50$ - $|E|=60$ - $|C|=40$ - $|M \cap E|=20$ - $|M \cap C|=15$ - $|E \cap C|=10$ - $|M \cap E \cap C|=5$ - Total students $=120$ 3. **Formula used:** To find the number of students studying at least one subject, use the Inclusion-Exclusion Principle: $$|M \cup E \cup C| = |M| + |E| + |C| - |M \cap E| - |M \cap C| - |E \cap C| + |M \cap E \cap C|$$ 4. **Calculate the number of students studying at least one subject:** $$|M \cup E \cup C| = 50 + 60 + 40 - 20 - 15 - 10 + 5 = 150 - 45 + 5 = 110$$ 5. **Find the number of students who study none of the three subjects:** Total students $=120$, so those who study none: $$120 - 110 = 10$$ 6. **Find the number of students who study exactly one subject:** Use the formula for exactly one set: $$|M| - |M \cap E| - |M \cap C| + |M \cap E \cap C| = 50 - 20 - 15 + 5 = 20$$ $$|E| - |M \cap E| - |E \cap C| + |M \cap E \cap C| = 60 - 20 - 10 + 5 = 35$$ $$|C| - |M \cap C| - |E \cap C| + |M \cap E \cap C| = 40 - 15 - 10 + 5 = 20$$ Sum exactly one subject: $$20 + 35 + 20 = 75$$ 7. **Find the number of students who study exactly two subjects:** Subtract those studying all three from each pair: $$|M \cap E| - |M \cap E \cap C| = 20 - 5 = 15$$ $$|M \cap C| - |M \cap E \cap C| = 15 - 5 = 10$$ $$|E \cap C| - |M \cap E \cap C| = 10 - 5 = 5$$ Sum exactly two subjects: $$15 + 10 + 5 = 30$$ **Summary:** - Students studying none: 10 - Students studying exactly one subject: 75 - Students studying exactly two subjects: 30 - Students studying all three subjects: 5