Question

(b) 100 students were asked whether they had taken
courses in any of the three subjects, Mathematics,
Computer Science and Statistics. The results were
given below:45 had taken Mathematics: 18 had taken
Mathematics and Computer Science: 38 had taken
Computer Science: 21 had taken Statistics: 9 had
taken Mathematics and Statistics; 4 had Computer
Sdence and Statistics and 23 had taken ne courses in
any of these subjects. Draw a Venn diagram that will
show the results of the Survey.​

Answer 1

Given :

100 students who had taken  courses in any of the three subjects:

1. Mathematics
2. Computer Science
3. Statistics

To do:

Draw a Venn diagram that will  show the results of the Survey.​

Solution:

Let us Denote the set of students as :

Mathematics-M

Computer science-C

and Statistics-S

Then as per given data

$M \cap C=18\\\\M \cap S=9\\\\C \cap S=4$

$n(M) =45, n(C)=38 \, ,\, n(S)=21 \, and \, n(U) =100$

We know that

$n( M\cup C\, \cup S)' = n(U) -n( M\cup C\, \cup S)\\\\Thus \\\\n( M\cup C\, \cup S)= n(U) -n( M\cup C\, \cup S)'\\\\=100-23=77$

We know that  

$n( M\cup C \cup S) =n(M) +n(C)+n(S)-n(M\cap C)-n(C\cap S)-n(S\cap M)-n( M\cap C \cap S)$

$77=45+38+21-18-9-4+ \, n( M \cap C \cap S)\\\\77=73+ n( M \cap C \cap S)\\\\Thus\\\\ n( M \cap C \cap S)=77-73=4$

Thus 4 students have taken  all three subjects

Now we can draw Van diagram as attached