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Difficulty: MediumOverlapping Sets and Inclusion-Exclusion Principle

A clinical study evaluated 150150 patient records for the presence of three specific health biomarkers: Biomarker X, Biomarker Y, and Biomarker Z. The study revealed that 7070 patients had Biomarker X, 6060 patients had Biomarker Y, and 5050 patients had Biomarker Z. Exactly 3535 patients had exactly two of these biomarkers, and exactly 1010 patients had all three biomarkers. How many of the patients evaluated had none of the three biomarkers?

  1. A
    5
  2. B
    15
  3. 25Answer
  4. D
    35
  5. E
    45

Answer

25 patients evaluated had none of the three biomarkers.
By the inclusion-exclusion principle for three sets, the total number of elements in at least one set is given by ∣X∪Y∪Z∣=∣X∣+∣Y∣+∣Z∣−(Elements in exactly two sets)−2(Elements in all three sets)|X \cup Y \cup Z| = |X| + |Y| + |Z| - (\text{Elements in exactly two sets}) - 2(\text{Elements in all three sets}). Substituting the given values yields 70+60+50−35−2(10)=180−35−20=12570 + 60 + 50 - 35 - 2(10) = 180 - 35 - 20 = 125. Since there are 150150 total patients evaluated, the number of patients with none of the three biomarkers is 150−125=25150 - 125 = 25.

Step-by-Step Solution

1
Sum the individual counts for all three biomarker groups.
Sum of individual sets=70+60+50=180\text{Sum of individual sets} = 70 + 60 + 50 = 180
This sum counts individuals with 1 biomarker once, individuals with 2 biomarkers twice, and individuals with 3 biomarkers three times.
2
Apply the inclusion-exclusion formula adapted for 'exactly two' and 'all three' overlapping subsets.
At least one biomarker=180−(Exactly two)−2×(All three)=180−35−2(10)=125\text{At least one biomarker} = 180 - (\text{Exactly two}) - 2 \times (\text{All three}) = 180 - 35 - 2(10) = 125
To count each person with at least one biomarker exactly once, we subtract the count of patients with exactly two biomarkers once, and the count of patients with all three biomarkers twice.
3
Subtract the number of patients with at least one biomarker from the total population of patients evaluated.
None=150−125=25\text{None} = 150 - 125 = 25
The total group consists of patients with at least one biomarker plus patients with none of the three biomarkers.

Key Concept

Three-Set Overlapping Sets and Inclusion-Exclusion Principle
Estimated Time:1m 45s
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Topics

Overlapping Sets and Inclusion-Exclusion Principle

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