Question

In a class of 200 students, 125 students have taken programming language course, 85 students have taken data structures course, 65 students have taken computer organization course, 50 students have taken both programming languages and data structures, 35 students have taken both programming languages and computer organization, 30 students have taken both data structures and computer organization, 15 students have taken all the three courses. How many students have not taken any of the three courses?

• A. 15
• B. 20
• C. 25
• D. 35

Answer 1

The correct option is C 25

Method I:

Using Venn diagram we can easily see that Number of students have not taken any of the three subjects = 25

Method II:
Let p = {programming language}, D = {Data structure}, & C = {computer organization}

$Givenp\left(P\right)=\frac{125}{200},p\left(D\right)=\frac{85}{200},p\left(C\right)=\frac{65}{200}$

$p(P\cap D)=\frac{50}{200},p(P\cap C)=\frac{35}{200}$

$p(D\cap C)=\frac{30}{200},p(P\cap D\cap C)=\frac{15}{200}$

Using Addition theorem,
$p(P\cup D\cup C)=p\left(P\right)+p\left(D\right)+p\left(C\right)-p(P\cap D)-p(D\cap C)-p(P\cap C)+p(P\cap D\cap C)$
$=\frac{7}{8}$

$\Rightarrow p(\overline{P}\cap \overline{D}\cap \overline{C})]=1-P(P\cup D\cup C)$

$=1-\frac{7}{8}=\frac{1}{8}$

Hence, Number of students who had not taken any of the three courses

$=\frac{1}{8}\times 200=25$