In a party, each person takes at least one beverage. There are three beverages in the party – tea, coffee and milk. Each beverage is consumed by 30 persons. Five persons, who take tea, also take milk. Ten persons, who take milk, also take coffee. However, no person takes tea and coffee both. How many persons are there in the party ?
90
85
80
75
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CISF-AC-EXE – 2023
– We are given: |T| = 30, |C| = 30, |M| = 30.
– |T ∩ M| = 5 (Persons who take tea and milk).
– |M ∩ C| = 10 (Persons who take milk and coffee).
– |T ∩ C| = 0 (No person takes tea and coffee both).
– Since no one takes tea and coffee together (|T ∩ C| = 0), it implies that no one takes tea, coffee, *and* milk together (|T ∩ C ∩ M| = 0).
– Each person takes at least one beverage, so the total number of persons in the party is the size of the union of the three sets: |T ∪ C ∪ M|.
– The Principle of Inclusion-Exclusion for three sets is:
|T ∪ C ∪ M| = |T| + |C| + |M| – (|T ∩ C| + |T ∩ M| + |C ∩ M|) + |T ∩ C ∩ M|
– Substitute the given values:
|T ∪ C ∪ M| = 30 + 30 + 30 – (0 + 5 + 10) + 0
|T ∪ C ∪ M| = 90 – 15 + 0
|T ∪ C ∪ M| = 75.
– Tea and Milk only: 5 – 0 = 5
– Milk and Coffee only: 10 – 0 = 10
– Tea and Coffee only: 0
People taking only one beverage:
– Tea only: |T| – (|T ∩ M| + |T ∩ C|) + |T ∩ C ∩ M| = 30 – (5 + 0) + 0 = 25
– Coffee only: |C| – (|C ∩ M| + |C ∩ T|) + |T ∩ C ∩ M| = 30 – (10 + 0) + 0 = 20
– Milk only: |M| – (|M ∩ T| + |M ∩ C|) + |T ∩ C ∩ M| = 30 – (5 + 10) + 0 = 15
Total persons = (Tea only) + (Coffee only) + (Milk only) + (Tea & Milk only) + (Milk & Coffee only) + (Tea & Coffee only) + (All three)
Total = 25 + 20 + 15 + 5 + 10 + 0 + 0 = 75.