Question

There is a group of $265$ persons who like either singing or dancing or painting.

In this group, $200$ like singing, $110$ like dancing and $55$ like a painting.

If $60$ persons like both singing and dancing, $30$ like both singing and painting and $10$ like all three activities, then the number of persons who like only dancing and painting is

• A. $10$
• B. $20$
• C. $30$
• D. $40$

Answer 1

The correct option is A

$10$

Explanation for the correct answer:

Compute the required number:

Let $D$ denote dancing,$P$ denote painting,$S$ denote singing

$\therefore$ $n\left(D\cup P\cup S\right)=265$

$n\left(S\right)=200,n\left(D\right)=110,n\left(P\right)=55,n\left(S\cap D\right)=60,n\left(S\cap P\right)=30,n\left(S\cap D\cap P\right)=10$

$\because$ $n(D\cup P\cup S)=n\left(D\right)+n\left(P\right)+n\left(S\right)-n(D\cap P)-n(P\cap S)-n(S\cap D)+n(D\cap P\cap S)$

Substitute all values

$265=200+110+55-n\left(D\cap P\right)-30-60+10$

$n\left(D\cap P\right)=20$

The number of people liking only dancing and painting $=\left(D\cap P\right)-n\left(S\cap D\cap P\right)$

$=20-10$

$=10$

Hence, $10$ people like only dancing and painting.

Hence, option $\left(A\right)$ is the correct answer.