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 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

Answer: (A)

$10$

Let $D$ denotes dancing, $P$ denotes painting and $S$ denotes singing.
$\therefore n(D\cup P\cup S)=265$
$n\left(S\right)=200,n\left(D\right)=110,n\left(P\right)=55,n(S\cap D)=60,n(S\cap P)=30$ and $n(D\cap P\cap S)=10$
$\because n(D\cup P\cup S)=n\left(D\right)+n\left(P\right)+n\left(S\right)-n(D\cap P)-n(P\cap S)-n(S\cap D)+n(D\cap P\cap S)$
$\therefore 265=110+55+200-n(D\cap P)-30-60+10$
$\Rightarrow 265=285-n(D\cap P)$
$\Rightarrow n(D\cap P)=20$
$\therefore$ Only person who like dancing and painting $=n(D\cap P)-n(D\cap P\cap S)$
$=20-10=10$