**Question:**

A $ 50 $ summer camp student is offered two sports: badminton and table tennis. There is $ 30 $ you register to play badminton, $ 28 $ you register to play table tennis and $ 10 $ you don’t register to play any game. How many friends are there: a) Register to play both subjects? B) Only register to play one subject?

**Answer**

The symbol $ X $ is the collection of students in the class. $ A, B $ are the set of students registered to play badminton and table tennis respectively. Thus, the set of students registering to play both subjects is $ A \cap B $. The set of students who register at least one subject is $ A \cup B $.

Clearly $ N (A \cup B) = 50-10 = 40 $ a) We have $ N ( A \cup B) = N (A) + N (B) -N (A \cap B) $

$ \Rightarrow N (A \cap B) = N (A) + N (B ) -N (A \cup B) = 30 + 28-40 = 18 $

So there are $ 18 $ students registering to play both subjects

b) The number of students who register to play only one subject is < br> $ N (A \cup B) -N (A \cap B) = 40-18 = 22 $