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Difficulty: MediumLogical Venn Diagrams

A state health department surveyed 800800 rural clinics regarding the availability of three essential facilities: Telemedicine (TT), Maternal Care Units (MM), and 24/724/7 Ambulance Services (AA).

The survey results revealed the following:
- 410410 clinics have Telemedicine.
- 460460 clinics have Maternal Care Units.
- 400400 clinics have 24/724/7 Ambulance Services.
- 220220 clinics have both Telemedicine and Maternal Care Units.
- 230230 clinics have both Maternal Care Units and Ambulance Services.
- 210210 clinics have both Telemedicine and Ambulance Services.
- 5050 clinics do not have any of these three facilities.

Based on the data provided, which of the following statements are correct? (Select all that apply)

  1. Exactly 140140 clinics have all three facilities.Answer
  2. B
    Exactly 660660 clinics have exactly two of the facilities.
  3. Exactly 370370 clinics have exactly one facility.Answer
  4. D
    Exactly 220220 clinics have Telemedicine and Maternal Care Units, but do not have Ambulance Services.

Answer

The correct statements are that exactly 140 clinics have all three facilities, and exactly 370 clinics have exactly one facility.
Based on the Venn diagram calculation, solving for the triple intersection yields exactly 140 clinics. Subsequently calculating the strictly single regions (Telemedicine only, Maternal Care only, and Ambulance only) yields 120, 150, and 100 respectively, which safely sum to exactly 370 clinics having only one facility.

Step-by-Step Solution

1
Determine the total number of clinics that have at least one facility.
Total clinics having at least one facility = 800−50=750800 - 50 = 750.
To apply the standard three-set inclusion-exclusion principle, the population must only include those inside the union of the three sets.
2
Calculate the number of clinics having all three facilities (T∩M∩AT \cap M \cap A).
Apply the formula: ∣T∪M∪A∣=∣T∣+∣M∣+∣A∣−∣T∩M∣−∣M∩A∣−∣T∩A∣+∣T∩M∩A∣|T \cup M \cup A| = |T| + |M| + |A| - |T \cap M| - |M \cap A| - |T \cap A| + |T \cap M \cap A|. Substituting the values gives: 750=410+460+400−220−230−210+∣T∩M∩A∣750 = 410 + 460 + 400 - 220 - 230 - 210 + |T \cap M \cap A|. Simplifying this yields 750=610+∣T∩M∩A∣750 = 610 + |T \cap M \cap A|, meaning ∣T∩M∩A∣=140|T \cap M \cap A| = 140.
This establishes the exact center of the Venn diagram, which is fundamentally required to find the exact values for all other independent regions.
3
Determine the number of clinics having exactly two facilities.
Exactly T and M: 220−140=80220 - 140 = 80. Exactly M and A: 230−140=90230 - 140 = 90. Exactly T and A: 210−140=70210 - 140 = 70. Total exactly two = 80+90+70=24080 + 90 + 70 = 240.
The pairwise intersections given in the problem include the clinics that have all three facilities. We must subtract the triple intersection from each pairwise intersection to avoid double counting.
4
Calculate the total number of clinics having exactly one facility.
Only T = 410−(80+70+140)=120410 - (80 + 70 + 140) = 120. Only M = 460−(80+90+140)=150460 - (80 + 90 + 140) = 150. Only A = 400−(70+90+140)=100400 - (70 + 90 + 140) = 100. Total exactly one = 120+150+100=370120 + 150 + 100 = 370.
To find the strictly single categories, we must subtract the 'exactly two' regions and the 'all three' region from the original total of each designated set.

Key Concept

Inclusion-Exclusion Principle and 3-Set Logical Venn Diagrams
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Topics

Logical Venn Diagrams

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