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Among 100 commuters surveyed, 50 use a bus, 40 use a train, and 25 use a bicycle. Additionally, 12 use both a bus and a train, 10 use both a bus and a bicycle, and 8 use both a train and a bicycle, while 5 use all three modes. How many commuters use exactly one of the three modes?
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Correct answer
C
Union = 50 + 40 + 25 − 12 − 10 − 8 + 5 = 90 (so 10 use none). The all-three group is 5. Exactly-two = (sum of pairwise counts) − 3 × (all three) = (12 + 10 + 8) − 15 = 15, because each pairwise count already includes the 5 who use all three. Exactly-one = union − exactly-two − exactly-three = 90 − 15 − 5 = 70.
The dominant false summit is reading the given pairwise figures as exactly-two (skipping the triple-overlap subtraction), which yields 55. The item stacks inclusion-exclusion for the union with the exactly-k partition.
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