Question

A survey on a sample of 25 new cars checked for three options—air-conditioning (A), radio (R), and power windows (P). Data: 15 had A; 2 had A&P but no R; 12 had R; 6 had A&R but no P; 11 had P; 4 had R&P (total); 3 had all three. How many cars had none of the options?

• A. 4
• B. 3
• C. 1
• D. 2

Hint:

Inclusion-exclusion is easiest when you first fix the "only" regions using totals and the given pairwise/three-way counts.

Answer 1

The Correct Option is D

Step 1: Translate to "only" groups. 
Given: \(|A|=15,\ |R|=12,\ |P|=11\). 
A&P only \(=2\), A&R only \(=6\); R&P total \(=4\) includes all three \(=3\) \(\Rightarrow\) R&P only \(=1\). 

Step 2: Find single-only counts. 
A only \(= 15 - (6+2+3) = 4\). 
R only \(= 12 - (6+1+3) = 2\). 
P only \(= 11 - (2+1+3) = 5\). 

Step 3: Sum "at least one". 
Singles: \(4+2+5 = 11\). 
Two-only: \(6+2+1 = 9\). 
All three: \(3\). 
Total with \(\ge 1\) option \(= 11+9+3 = 23\). 

Step 4: Cars with none. 
Total \(=25 \Rightarrow\) None \(= 25 - 23 = \boxed{2}\).