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 cars had to air-condition.
2 cars had air-conditioning and power windows but no radios.
12 cars had power windows.
6 cars had air-conditioning and radio but no power windows.
11 cars had a radio.
4 cars had radio and power windows.
3 cars 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

Hint: Using, the data given draw the Venn-diagram. Now, calculate the number of cars that had any of the options from radio, air-conditioning, and power windows. It is given that there are a total of 25 cars. Then, solve it further and calculate the number of cars that had none of the options.

Complete step-by-step solution:
According to the question, it is given that there was 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.
15 cars had to air-condition. This doesn’t mean that it only had to air-condition. It can also have a radio, or power-windows, or both.
2 cars had air-conditioning and power windows but no radios.
12 cars had power windows. This doesn’t mean that it only had power windows. It can also have a radio, or air-conditioning, or both.
6 cars had air-conditioning and radio but no power windows.
11 cars had a radio. This doesn’t mean that it only had a radio. It can also have power-windows, or air-conditioning, or both.
4 cars had radio and power windows. It may or may not have to air-condition.
3 cars had all three options.
Now, on drawing Venn diagram, we get

It is given that 15 cars had to air-condition.
From the Venn-diagram,
The number of cars that had only air-conditioning = \[15-\left( 6+3+2 \right)=15-11=4\] …………………………………..(1)
It is also given that 11 cars had power windows.
From the Venn-diagram,
The number of cars that had only power-windows = \[11-\left( 2+3+1 \right)=11-6=5\] …………………………………..(2)
It is also given that 12 cars had a radio.
From the Venn-diagram,
The number of cars that had only radio = \[12-\left( 6+3+1 \right)=12-10=2\] …………………………………..(3)
From the Venn-diagram, we can calculate the total number of cars that had any of the options from radio, power-windows, or air-conditioning.
The total number of cars that had any of the options from radio, power-windows, or air-conditioning = \[4+6+3+2+1+5+2=23\] …………………………………(4)
It means 23 cars had any of the options from radio, power-windows, or air-conditioning.
It is given that there are a total of 25 cars.
Therefore, the number of cars that had none of the options = \[25-23=2\].
Hence, the correct option is (B).

Note: In this question, one might assume that 15 cars had only air-conditioning. This is wrong because it is not mentioned strictly that 15 cars had only air-conditioning. It is mentioned that 15 cars had to air-condition. So, they can have other options from radio and power-windows as well.