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Difficulty: HardSets, Set Operations, and Venn Diagrams

In a medical study of 120120 hospital outpatients, 6868 are prescribed Green tea extract (GG), 5252 are prescribed Herbal infusion (HH), and 4444 are prescribed Chamomile syrup (CC). Furthermore, 2424 patients receive both Green tea extract and Herbal infusion, 1818 receive both Green tea extract and Chamomile syrup, 1616 receive both Herbal infusion and Chamomile syrup, while 88 receive all three prescriptions. How many patients receive at most one of these three prescriptions?

  1. A
    7272
  2. 7878Answer
  3. C
    5454
  4. D
    112112

Answer

The number of patients who receive at most one of the three prescriptions is 7878.
To find the number of patients who receive at most one prescription, we must calculate the number of patients who receive exactly one prescription as well as those who receive none (the complement of the union relative to the universal set of 120 outpatients). The number receiving only Green tea is 3434, only Herbal infusion is 2020, and only Chamomile syrup is 1818, giving 7272 patients taking exactly one prescription. The total taking at least one prescription is 114114, leaving 120−114=6120 - 114 = 6 patients receiving none. Adding these together yields 72+6=7872 + 6 = 78.

Step-by-Step Solution

1
Calculate the number of patients in each exclusive intersection region.
Patients receiving all three n(G∩H∩C)=8n(G \cap H \cap C) = 8. Patients receiving only G∩H=24−8=16G \cap H = 24 - 8 = 16. Patients receiving only G∩C=18−8=10G \cap C = 18 - 8 = 10. Patients receiving only H∩C=16−8=8H \cap C = 16 - 8 = 8.
Subtracting the triple intersection isolates the regions representing patients taking exactly two prescriptions.
2
Calculate the number of patients receiving exactly one prescription.
n(G only)=68−(16+10+8)=34n(G \text{ only}) = 68 - (16 + 10 + 8) = 34. n(H only)=52−(16+8+8)=20n(H \text{ only}) = 52 - (16 + 8 + 8) = 20. n(C only)=44−(10+8+8)=18n(C \text{ only}) = 44 - (10 + 8 + 8) = 18. Total receiving exactly one =34+20+18=72= 34 + 20 + 18 = 72.
Each set total is reduced by its overlapping intersection regions to yield the single-prescription regions.
3
Determine the number of patients receiving at least one prescription using the inclusion-exclusion principle.
n(G∪H∪C)=(68+52+44)−(24+18+16)+8=164−58+8=114n(G \cup H \cup C) = (68 + 52 + 44) - (24 + 18 + 16) + 8 = 164 - 58 + 8 = 114.
The principle of inclusion-exclusion accounts for double-counted pairwise intersections and triple-counted intersections.
4
Calculate the number of patients receiving none of the prescriptions.
n((G∪H∪C)′)=120−114=6n((G \cup H \cup C)') = 120 - 114 = 6.
Subtracting the total taking at least one prescription from the universal set gives the complement count.
5
Sum the patients receiving exactly one prescription and those receiving zero prescriptions.
Total taking at most one =72+6=78= 72 + 6 = 78.
'At most one' encompasses both 'zero prescriptions' and 'exactly one prescription'.

Key Concept

Three-set principle of inclusion-exclusion and complementary sets within a universal set
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Topics

Sets, Set Operations, and Venn Diagrams

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