UPSC NDA I 2018 Mathematics Solved Paper

Concept:Use the principle of inclusion-exclusion for sets to find the number of students who like both music and dance.Explanation:Let $M$ be the set of students who like music and $D$ be the set of students who like dance.Given total students $|X| = 850$, $|M| = 680$, $|D| = 215$.We need $|M \cap D|$, the number who like both.The maximum possible union size is the total number of students, i.e., $|M \cup D| \le 850$.Using the formula:$|M \cup D| = |M| + |D| - |M \cap D|$To find the least possible $|M \cap D|$, we take the largest possible union i.e., $850$.Then, $850 = 680 + 215 - |M \cap D|$So, $|M \cap D| = 680 + 215 - 850 = 895 - 850 = 45$.Thus, at least 45 students like both.Answer:45