Question

In a survey of $60$ people, it was found that $25$ people read newspaper $H,26$ read newspaper $T,26$ read newspaper $I,9$ read both $H\&I,11$ read both $H\&T,8$ read both $T\&I,3$ read all three newspapers, then

• A. $26$ People read exactly one newspaper
• B. $30$ People read exactly one newspaper
• C. $19$ People read exactly two of the three newspaper
• D. $28$ People read exactly two of the three newspaper

Answer 1

Answer: (B), (C)

$19$ People read exactly two of the three newspaper

Let: $H=$ Set of people who read newspaper $H$
$T=$ Set of people who read newspaper $T$
$I=$ Set of people who read newspaper $I$
$\Rightarrow n\left(U\right)=60;n\left(H\right)=25;n\left(T\right)=26;$
$n\left(I\right)=26;n(H\cap I)=9;n(H\cap T)=11;$ and $n(T\cap I)=8,n(H\cap T\cap I)=3$
Now, let's create a venn diagram for the same and label them as shown:

Now, according to the diagram and the data given we can write:
$g=n(H\cap T\cap I)=3$
$\Rightarrow g=3$
$n(H\cap T)=e+g=11\Rightarrow e=8$

$n(H\cap I)=d+g=9\Rightarrow d=6$

$n(T\cap I)=f+g=8\Rightarrow f=5$

Now, $n\left(H\right)=a+e+g+d=25$
$\Rightarrow a+8+3+6=25\Rightarrow a=8$

$n\left(T\right)=b+e+g+f=26$
$\Rightarrow b+8+3+5=26\Rightarrow b=10$

$n\left(I\right)=c+f+g+d=25$
$\Rightarrow c+5+3+6=26\Rightarrow c=12$

Now, number of people who read exactly one newspaper $=a+b+c=8+10+12=30$

Number of people who read exactly two of three newspaper:
$=d+e+f=6+8+5=19$