Question

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 1

Let C denote the set of people who like coffee, and T denote the set of people who like tea
$n(C\cup T)=70,n\left(C\right)=37,n\left(T\right)=52$

We know that:
$n(C\cup T)=n\left(C\right)+n\left(T\right)-n(C\cap T)$
$\therefore 70=37+52-n(C\cap T)$
$\Rightarrow 70=89-n(C\cap T)$
$\Rightarrow n(C\cap T)=89-70=19$

Thus, 19 people like both coffee and tea