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. Following were the observation of the survey:
I. 15 had air conditioning
II. 2 had air conditioning and power windows but no radios
III. 12 had radio
IV. 6 had air conditioning and radio but no power windows
V. 11 had power windows
VI. 4 had radio and power windows
VII. 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

Hint:

When dealing with set problems, use the principle of inclusion-exclusion to account for overlapping sets.

Answer 1

The Correct Option is B

We are given the following:
- Total cars = 25
- Cars with air conditioning = 15
- Cars with radio = 12
- Cars with power windows = 11
- Cars with all three options = 3
We use the principle of inclusion-exclusion to find the number of cars with at least one of the options: \[ \text{Total with at least one option} = \text{air conditioning} + \text{radio} + \text{power windows} - (\text{pairs of options}) + \text{all three options} \] Thus: \[ \text{Total with at least one option} = 15 + 12 + 11 - 2 - 6 - 4 + 3 = 29 - 12 = 17 \] Thus, the number of cars that had none of the options is: \[ 25 - 17 = 3 \]