A survey of 50 cars was conducted for the features air conditioning, radio,…
2016
A survey of 50 cars was conducted for the features air conditioning, radio, and power windows. The following information was found:
30 cars had air conditioning.
4 cars had air conditioning and power windows but no radio.
21 cars had radio.
12 cars had radio and air conditioning but no power windows.
20 cars had power windows.
1 car had radio and power windows but no air conditioning.
6 cars had all three features.
How many cars had none of these three features?
Answer: B. 8 — ConceptFor three sets, inclusion–exclusion gives the size of their union by adding the three single-set totals, subtracting the three pairwise intersections,…
- A.
6
- B.
8
- C.
10
- D.
4
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Show answer & explanation
Correct answer: B
Concept
For three sets, inclusion–exclusion gives the size of their union by adding the three single-set totals, subtracting the three pairwise intersections, and then adding the triple intersection once.
A count stated as “in A and B but not C” is an exclusive two-set region. Add the triple-overlap count to it before using it as an inclusive pairwise intersection.
Application
Let A, R, and P denote cars with air conditioning, radio, and power windows. The inclusive pairwise intersections are |A∩R| = 12 + 6 = 18, |A∩P| = 4 + 6 = 10, and |R∩P| = 1 + 6 = 7.
Apply inclusion–exclusion: |A∪R∪P| = 30 + 21 + 20 − 18 − 10 − 7 + 6.
Evaluate the union: |A∪R∪P| = 42. Thus, 42 cars have at least one of the three features.
Take the complement in the surveyed total: 50 − 42 = 8 cars have none of the three features.
Cross-check
Compute the seven disjoint regions and verify that their sum equals the union:
Region | Count |
|---|---|
Air conditioning only | 30 − 12 − 4 − 6 = 8 |
Radio only | 21 − 12 − 1 − 6 = 2 |
Power windows only | 20 − 4 − 1 − 6 = 9 |
Exactly A and R | 12 |
Exactly A and P | 4 |
Exactly R and P | 1 |
All three | 6 |
The disjoint-region sum is 8 + 2 + 9 + 12 + 4 + 1 + 6 = 42, so the complement is again 8.
Result
8 cars had none of the three features.