In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?

In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?

Answer
Let C and T represent the set of people who like coffee and tea respectively.
Number of person likes at least one of the two drinks, n(C ∪ T) = 70.
Number of person like coffee, n(C) = 37
Number of person likes tea, n(T) = 52
Use the formula
n(C ∪ T) = n(C) + n(T) - n(C ∩ T)
Plug the values, we get
70 = 37 + 52 - n(C ∩ T)
⇒ 70 = 89 - n(C ∩ T)
⇒ n(C ∩ T) = 89 - 70 = 19
Thus, 19 people like both coffee and tea.