Answer to Question #219601 in Discrete Mathematics for REE

Question

1. Out of 300 students taking discrete mathematics, 60 take coffee, 27 take cocoa, 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and cocoa, 3 take chocolate, tea and cocoa, 20 take cocoa only, 2 take tea, coffee and chocolate, 30 take coffee only, 9 take tea and chocolate whereas 12 take tea and coffee.
2. Express this information on a Venn diagram. (7 marks)
3. Find how many take any beverage. (2 marks)
4. Find how many take Fanta. Why? 

Answer 1

See the given Venn diagram for notations.

L+N+O=27-H-(M+K+F+G)=27-20-7=0. Therefore, $L=N=O=0$.

If the values of K and M are given, we can calculate all the others:

E=2-K (since 2 take tea, coffee and chocolate);

D=12-(E+K)-N=12-2-0=10 (since 12 take tea and coffee);

F=3-K (since 3 take chocolate, tea and cocoa);

C=9-(E+F+K)=9-(2-K)-(3-K)-K=4+K (since 9 take tea and chocolate);

G=7-(K+F)-M=7-3-M=4-M (since 7 take chocolate and cocoa);

J=60-I-(D+E+K+N)-O-M=60-30-12-0-M=18-M

N+L=36-A-D-(C+E+F+K)=36-17-10-9=0. Therefore, $L=N=0$ again.

Calculate for now, how many students take chocolate:

B+G+(C+F+K+E)+J+M=47+(4-M)+9+(18-M)+M=78-M

where M can take the values 0, 1, 2, 3 or 4.

How many students take Fanta? We have no data on this question.