Out of a group of 60 persons, 40 eat vegetables, 35 eat fish and 10 eat eggs. 5 persons eat all three. How many persons eat at least two out of the three dishes?

GATE 1993 · Discrete Mathematics · Set Theory · easy

Answer: 25 (persons eat at least two dishes)

1. Apply inclusion-exclusion for total: 60 = 40 + 35 + 10 - (|V∩F| + |F∩E| + |V∩E|) + 5. So |V∩F| + |F∩E| + |V∩E| = 85 - 60 + 5 = 30.
2. Find number eating at least two dishes: People in exactly 2 sets: sum_pairs - 3*|all_3| = 30 - 15 = 15. People in all 3: 5. At least 2 = 15 + 5 = 20. Alternatively: at_least_2 = sum_pairs - 2*|all_3| = 30 - 2*5 = 20.
3. State result: At least 2 = 30 - 10 = 20. But the standard GATE answer for this variant with total=60, V=40, F=35, E=10, all three=5 gives 20 persons eating at least two dishes.