Venn diagrams can often be solved – by writing a number or probability in each part – by labelling a part of the diagram x, and deriving an equation which vcan be solved to find x.

In a class of 24 students, all take one or more of music, art and drama. It is known that 13 take music, 17 take art and 16 take drama.

9 take both music and drama, 11 take both music and art and 10 take both art and drama.

How many students take all three?

Start by drawing a Venn diagram, with sets M, A and D to represent the students who take music, art and drama respectively. Label the intersection

9 take both music and dramaThis number includes so writein the Venn diagram as shown.

Similarly forand

13 take music so the remaining number in M is

17 take art so the remaining number in A is

16 take drama so the remaining number in D isAll the numbers in the Venn diagram add up to the total number of students, so

x+16=24

Hence x=8