Question

Consider the following statements :
1. The number of students who had taken only one subject is equal to the number of students who had taken only two subjects.
2. The number of students who had taken at least two subjects is four times the number of students who had taken all the three subjects. Which of the above statements is/are correct ?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is B 2 only

$n\left(M\right)=15$

$n\left(P\right)=12$

$n\left(C\right)=11$

$n(M\cap P)=9$

$n(M\cap C)=5$

$n(P\cap C)=4$

$n(P\cap M\cap C)=3$

We can see from the diagram,

Number of students who only took one subject $\left(y\right)=\left[n\right(M)-n(M\cap C)-n(M\cap P)+n(M\cap P\cap C\left)\right]+\left[n\right(P)-n(M\cap P)-n(P\cap C)+n(M\cap P\cap C\left)\right]+\left[n\right(C)-n(M\cap C)-n(P\cap C)+n(P\cap C\cap M\left)\right]$

$=[15-9-5+3]+[12-9-4+3]+[11-5-4+3]=4+2+5=11$

Number of students who have taken only two subjects

$\left(x\right)=n(M\cap C)+n(M\cap P)+n(P\cap C)-3n(M\cap C\cap P)$

$=5+9+4-9=9$

No. of students who have taken at least two subjects

$\left(z\right)=x+n(P\cap C\cap M)=9+3=12$

Therefore we see that only statement 2 is correct.

Hence, B is correct.