Question

A group of $123$ workers went to a canteen for cold drink, ice-cream and tea. $42$ workers took ice-cream, $36$ took tea and $30$ took cold drink. $15$ workers purchased ice-cream and tea, $10$ purchased ice-cream and cold drink, and $4$ purchased cold drink and tea but not ice-cream. Then the number of workers who did not purchase anything is

• A. $60$
• B. $79$
• C. $44$
• D. $40$

Answer 1

The correct option is C $44$

Let $C,I$ and $T$ denote the set of workers who purchased cold drink, ice-cream and tea respectively.
$n\left(U\right)=123$
$n\left(I\right)=42,n\left(T\right)=36,n\left(C\right)=30$
$n(I\cap T)=15,n(I\cap C)=10$
$n(C\cap T\cap I^{\prime })=4$
$\Rightarrow n(C\cap T\cap I^{\prime })=4=n(C\cap T)-n(C\cap T\cap I)$

Number of workers who did not purchase anything
$=n(C\cup T\cup I{)}^{\prime }$
$=n\left(U\right)-n(C\cup T\cup I)$
$=n\left(U\right)-\left[n\right(I)+n(T)+n(C)-n(I\cap T)-n(I\cap C)-n(C\cap T)+n(C\cap T\cap I\left)\right]$
$=123-[42+36+30-15-10-4]$
$\therefore n(C\cup T\cup I{)}^{\prime }=44$