## Class Distinction

#### Problem

Mr. Venn draws two large overlapping circles on the floor of the sports hall and labels them B and R. He asks all those students with brown hair to stand in the B circle and those that are right handed to stand in the R circle; if they have both brown hair and are right handed, they need to stand in the region where the two circles overlap.

When they return to the classroom he asks his class of thirty two students how many have brown hair: twenty seven put their hands up. He then asks how many students are right handed: twenty four raise their hands.

What is the minimum number of students that stood in the overlap?

#### Solution

From the information, we can deduce that 32 24 = 8 students are left handed.

As we are trying the minimise the number of students with brown hair that are right handed, we would like to maximise the number of brown haired students that are left handed.

We begin by assigning all eight left handed students to the region that is in B but not in R:

As there are twenty seven students in total with brown hair, we know that 27 8 = 19 students that have brown hair and are right handed. This is represented by the intersection of the B and R circles. For completion we shall fill in the other values.

That is, the minimum number students that stood in the overlap is 19.

What is the maximum number of right handed students with brown hair?

In terms of being right/left handed and having/not having brown hair, what do the values 5 and 0 on the diagram represent?