Question

In a survey of 25 students, it was found that 15 had taken mathematics, 12 had taken physics and 11 had taken chemistry, 5 had taken mathematics and chemistry, 9 had taken mathematics and physics, 4 had taken physics and chemistry and 3 had taken all the three subjects.

Find the number of students that had taken none of the subjects.

• A. $0$
• B. $2$
• C. $3$
• D. $5$

Answer 1

The correct option is B $2$

Let M=Set of the student who taken Mathematics only

C=Set of the student who taken chemistry only.

P=Set of the student who taken Physics only.

$\therefore n\left(M\right)=15,n\left(P\right)=12,n\left(C\right)=11,n(M\cap C)=5,n(M\cap P)=9,n(P\cap C)=4,n(M\cap P\cap C)=3$

$n(M\cup P\cup C)=n\left(M\right)+n\left(P\right)+n\left(C\right)-n(P\cap M)-n(P\cap C)-n(C\cap M)+n(M\cap C\cap P)$

$=15+12+11-9-4-5+3=23$

The number of students that had taken none of the subjects

$n(M\text{'}\cap P\text{'}\cap C\text{'})=n(M\cup P\cup C)\text{'}=25-n(M\cup P\cup C)\Rightarrow 25-23=2$