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 ?

To find the total number of persons in the party, we use the Principle of Inclusion-Exclusion for three sets: Tea (T), Coffee (C), and Milk (M).

The formula for the total number of unique elements is:
Total = n(T) + n(C) + n(M) - [n(T ∩ C) + n(C ∩ M) + n(M ∩ T)] + n(T ∩ C ∩ M)

From the question, we have:

• n(T) = 30, n(C) = 30, n(M) = 30
• n(T ∩ M) = 5
• n(M ∩ C) = 10
• n(T ∩ C) = 0 (No person takes both tea and coffee)
• n(T ∩ C ∩ M) = 0 (Since n(T ∩ C) is 0, the intersection of all three must also be 0)

Substituting the values into the formula:
Total = 30 + 30 + 30 - (0 + 10 + 5) + 0
Total = 90 - 15 = 75

Therefore, there are 75 persons in the party.