Question

A survey on a sample of $25$ new cars being sold at a local auto dealer was conducted to see which of the three popular options - air-conditioning, radio and power windows - were already installed.
The survey found:
$15$ had air-conditioning
$2$ had air-conditioning and power windows but no radios.
$12$ had power windows
$6$ had air-conditioning and radio but no power windows.
$11$ had radio.
$4$ had radio and power windows.
$3$ had all three options.
What is the number of cars that had none of the options?

• A. $4$
• B. $3$
• C. $1$
• D. $2$

Answer 1

The correct option is D $2$

$n(A\cup R\cup P)=n\left(A\right)+n\left(R\right)+n\left(P\right)-n(A\cap R)-n(R\cap P)-n(P\cap A)+n(A\cap R\cap P)$

$2=n(A\cap P)-n(A\cap P\cap R)$

$\Rightarrow n(A\cap P)=2+3=5$

$6=n(A\cap R)-n(A\cap P\cap R)$

$\Rightarrow n(A\cap R)=6+3=9$

$n(A\cup R\cup P)=n\left(A\right)+n\left(R\right)+n\left(P\right)-n(A\cap R)-n(R\cap P)-n(P\cap A)+n(A\cap R\cap P)$

$=15+11+12-9-5-4+3$

So answer $=25-23=2$