\nLet C, H, and W be the sets of students who like cartoon, horror, and war movies, respectively. We are given |C| = 65%, |H| = 70%, and |W| = 75%. We want to find the minimum value of |C \u2229 H \u2229 W|.
\nLet’s consider the students who *dislike* each type of movie.
\nPercentage disliking C = 100% – 65% = 35%.
\nPercentage disliking H = 100% – 70% = 30%.
\nPercentage disliking W = 100% – 75% = 25%.
\nA 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 \u2229 H \u2229 W) is the complement of the set of students disliking at least one type (Dislike C U Dislike H U Dislike W).
\n|C \u2229 H \u2229 W| = 100% – |Dislike C U Dislike H U Dislike W|.
\nTo minimize |C \u2229 H \u2229 W|, we need to maximize |Dislike C U Dislike H U Dislike W|.
\nThe maximum possible value of the union of three sets is the sum of their individual sizes (if they are disjoint).
\nMax |Dislike C U Dislike H U Dislike W| <= |Dislike C| + |Dislike H| + |Dislike W| = 35 + 30 + 25 = 90%.\nIf 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).\nUsing the inclusion-exclusion principle for intersection:\n|C \u2229 H \u2229 W| >= |C| + |H| + |W| – 2 * 100% (since the maximum size of the total set is 100%)
\n|C \u2229 H \u2229 W| >= 65 + 70 + 75 – 200 = 210 – 200 = 10%.
\nThis formula gives the minimum possible intersection. Since we showed that 10% is achievable (when the dislike sets are disjoint), the minimum percentage is 10%.
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