Question

In a random survey 250 people participated. Out of 250 people who took part in the survey, 40 people have listened to Pink Floyd. 30 people have listened to metallica and 20 people have listened to John Denver. If 10 people have listened to all three then find the no. of people who have listened only Pink Floyd.

• A. 10
• B. 20
• C. 30
• D. 25

Answer 1

The correct option is C

30

Let the no. of people who have listened to Pink Floyd be n(P) = 40
Similarly, the no. of people who have listened to Metallica be n(M) = 30
and the no. of people who have listened to John Denver be = n(J) = 20
Also given that the no. of people who listen to all three i.e. $n(P\cap M\cap J)=10$
We have to find the no. of people who have listened to only Pink floyd.
Let that no. be x.
Now, from Demorgan's second law, we know that
$P-(M\cup J)=(P–M)\cap (P–J)$
$n(P-(M\cup J\left)\right)=n(P–M)+n(P–J)–n\left(\right(P–M)\cup (P–J\left)\right)$
$\Rightarrow n(P-(M\cup J\left)\right)=30+30-30$
$\Rightarrow n(P-(M\cup J\left)\right)=30$

Thus, 30 people listen to only Pink Floyd.