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Difficulty: MediumSet Theory Concepts and Venn Diagrams

A survey of 300300 urban commuters evaluated their usage of three transit services: the Bus (BB), the Commuter Rail (RR), and the Express Ferry (FF). The survey revealed the following information:

- 160160 commuters use the Bus.
- 140140 commuters use the Commuter Rail.
- 110110 commuters use the Express Ferry.
- 6060 commuters use both the Bus and the Commuter Rail.
- 4545 commuters use both the Commuter Rail and the Express Ferry.
- 5050 commuters use both the Bus and the Express Ferry.
- 2525 commuters use all three transit services.

How many of the surveyed commuters use none of these three transit services?

  1. 20Answer
  2. B
    45
  3. C
    70
  4. D
    280
  5. E
    145

Answer

20 commuters use none of the three transit services.
Using the inclusion-exclusion principle for three overlapping sets, the total number of commuters using at least one of the transit services is 160+140+110−(60+45+50)+25=280160 + 140 + 110 - (60 + 45 + 50) + 25 = 280. Subtracting this value from the total surveyed group of 300300 commuters gives 300−280=20300 - 280 = 20 commuters who use none of the three services.

Step-by-Step Solution

1
Apply the Principle of Inclusion-Exclusion for three sets to find the total number of commuters who use at least one transit service, ∣B∪R∪F∣|B \cup R \cup F|.
∣B∪R∪F∣=∣B∣+∣R∣+∣F∣−(∣B∩R∣+∣R∩F∣+∣F∩B∣)+∣B∩R∩F∣|B \cup R \cup F| = |B| + |R| + |F| - (|B \cap R| + |R \cap F| + |F \cap B|) + |B \cap R \cap F|
Simply adding set sizes double-counts elements in pairwise intersections and triple-counts elements in all three sets.
2
Substitute the given numerical values into the inclusion-exclusion formula.
∣B∪R∪F∣=160+140+110−(60+45+50)+25=410−155+25=280|B \cup R \cup F| = 160 + 140 + 110 - (60 + 45 + 50) + 25 = 410 - 155 + 25 = 280
Combining the sums and differences yields the exact count of commuters using at least one mode of transit.
3
Subtract the number of commuters using at least one service from the total surveyed population to find those using none.
None=300−280=20\text{None} = 300 - 280 = 20
The universe of surveyed commuters consists of those using at least one service plus those using none.

Key Concept

Principle of Inclusion-Exclusion for Three Sets
Estimated Time:1m 30s
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Topics

Set Theory Concepts and Venn Diagrams

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