A survey of 50 cars was conducted for the options air conditioning, radio and…
2016
A survey of 50 cars was conducted for the options air conditioning, radio and power windows. The following information was found:
(a) 30 cars had air conditioning
(b) 4 cars had air conditioning and power windows but no radio
(c) 21 cars had radio
(d) 12 cars had radio and air conditioning but no power windows
(e) 20 cars had power windows
(f) 1 car had radio and power windows but no air conditioning
(g) 6 cars had all three options
How many cars had none of these options?
- A.
6
- B.
8
- C.
10
- D.
4
Attempted by 209 students.
Show answer & explanation
Correct answer: B
Interpretation of the given data:
Air conditioning: 30 cars
Air conditioning and power windows but no radio: 4 cars
Radio: 21 cars
Radio and air conditioning but no power windows: 12 cars
Power windows: 20 cars
Radio and power windows but no air conditioning: 1 car
All three options: 6 cars
Note: The given pair counts are the numbers that exclude the third option. To use inclusion–exclusion, first compute the total counts for each pair including the triple.
Air conditioning and power windows (total) = 4 + 6 = 10
Radio and air conditioning (total) = 12 + 6 = 18
Radio and power windows (total) = 1 + 6 = 7
Use the inclusion–exclusion principle:
Number with at least one option = |AC| + |Radio| + |Power| − |AC∩Radio| − |AC∩Power| − |Radio∩Power| + |all three|
Substitute the numbers: 30 + 21 + 20 − 18 − 10 − 7 + 6 = 42
So 42 cars have at least one of the options. Therefore, cars with none = 50 − 42 = 8.