Concept: Principle of Inclusion-Exclusion for Three Sets.
Formula:$$|F \cup H \cup C| = |F| + |H| + |C| - (|F \cap H| + |H \cap C| + |F \cap C|) + |F \cap H \cap C|$$
Solution:- Total players \( |F \cup H \cup C| = 100 \)
- Football: \( |F| = 70 \), Hockey: \( |H| = 50 \), Cricket: \( |C| = 55 \)
- Two-set overlaps: Football & Hockey \( |F \cap H| = 25 \), Hockey & Cricket \( |H \cap C| = 30 \)
- Triple overlap (all three games): \( |F \cap H \cap C| = 20 \)
- Let \( x = |F \cap C| \) be the number of players who play both football and cricket.
Substituting into the Inclusion-Exclusion equation:
$$100 = 70 + 50 + 55 - (25 + 30 + x) + 20$$
$$100 = 175 - 55 - x + 20$$
$$100 = 140 - x$$
$$x = 140 - 100 = 40$$
Therefore, exactly
40 players play both football and cricket.
Why other options are incorrect:- Option A (25), Option B (30), Option C (35): Incorrect values resulting from misapplying the set intersection formula or erroneously omitting/double-counting the triple intersection set of 20 players.
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