65% students in a class like cartoon movies, 70% like horror movies, and 75% like war movies. What is the smallest percent of students liking all the three types of movies ?
10%
25%
30%
5%
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CAPF – 2015
Let’s consider the students who *dislike* each type of movie.
Percentage disliking C = 100% – 65% = 35%.
Percentage disliking H = 100% – 70% = 30%.
Percentage disliking W = 100% – 75% = 25%.
A student who likes all three types of movies is a student who does *not* dislike any of the three types. The set of students liking all three (C ∩ H ∩ W) is the complement of the set of students disliking at least one type (Dislike C U Dislike H U Dislike W).
|C ∩ H ∩ W| = 100% – |Dislike C U Dislike H U Dislike W|.
To minimize |C ∩ H ∩ W|, we need to maximize |Dislike C U Dislike H U Dislike W|.
The maximum possible value of the union of three sets is the sum of their individual sizes (if they are disjoint).
Max |Dislike C U Dislike H U Dislike W| <= |Dislike C| + |Dislike H| + |Dislike W| = 35 + 30 + 25 = 90%. If these dislike sets are disjoint, then 90% of students dislike at least one movie type. The remaining 100% - 90% = 10% must therefore like all three. This scenario is possible (e.g., different groups of students exclusively disliking one type). Using the inclusion-exclusion principle for intersection: |C ∩ H ∩ W| >= |C| + |H| + |W| – 2 * 100% (since the maximum size of the total set is 100%)
|C ∩ H ∩ W| >= 65 + 70 + 75 – 200 = 210 – 200 = 10%.
This formula gives the minimum possible intersection. Since we showed that 10% is achievable (when the dislike sets are disjoint), the minimum percentage is 10%.