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Difficulty: MediumOverlapping Sets and Venn Diagrams

In a biotechnology laboratory with 200200 research scientists, a survey was conducted regarding the use of three analytical instruments: Mass Spectrometers, NMR Spectrometers, and X-ray Diffractometers. The survey revealed that 9595 scientists use Mass Spectrometers, 8080 use NMR Spectrometers, and 7575 use X-ray Diffractometers. Exactly 2020 scientists use only Mass Spectrometers and NMR Spectrometers, exactly 1515 use only NMR Spectrometers and X-ray Diffractometers, and exactly 1010 use only Mass Spectrometers and X-ray Diffractometers. If 1515 scientists use all three instruments, how many scientists do not use any of these three instruments?

  1. A
    1010
  2. B
    1515
  3. 2525Answer
  4. D
    3535
  5. E
    5050

Answer

25 scientists do not use any of the three instruments.
The total number of scientists using at least one instrument is calculated by summing all disjoint regions: the scientists using only one instrument (50+30+35=11550 + 30 + 35 = 115), the scientists using exactly two instruments (20+15+10=4520 + 15 + 10 = 45), and the scientists using all three instruments (1515). This yields 115+45+15=175115 + 45 + 15 = 175 scientists who use at least one instrument. Subtracting this from the total sample of 200200 scientists gives 200−175=25200 - 175 = 25 scientists who use none of the three instruments.

Step-by-Step Solution

1
Identify the number of scientists in each distinct overlapping region
Regions with exactly two instruments are given as 20, 15, and 10. The region with all three instruments is 15.
To find the number of single-instrument users without double-counting, all overlapping regions must be isolated.
2
Calculate the number of scientists using only one instrument
Mass Spectrometers only: 95−(20+10+15)=5095 - (20 + 10 + 15) = 50. NMR Spectrometers only: 80−(20+15+15)=3080 - (20 + 15 + 15) = 30. X-ray Diffractometers only: 75−(10+15+15)=3575 - (10 + 15 + 15) = 35.
Subtracting all shared regions from each set's total yields the exclusive users of that single instrument.
3
Sum all mutually exclusive regions to find the total number of scientists using at least one instrument
Total using at least one = (Only 1 instrument) + (Only 2 instruments) + (All 3 instruments) = (50+30+35)+(20+15+10)+15=115+45+15=175(50 + 30 + 35) + (20 + 15 + 10) + 15 = 115 + 45 + 15 = 175.
The sum of disjoint regions equals the cardinality of the union of all three sets.
4
Subtract the number of scientists using at least one instrument from the total surveyed scientists
Scientists using none = 200−175=25200 - 175 = 25.
The set of scientists using none of the instruments is the complement of the union set.

Key Concept

3-Set Venn Diagram Inclusion-Exclusion Principle

Alternative Method

Using the 3-Set Inclusion-Exclusion formula: Total = |A| + |B| + |C| - (|A∩B| + |B∩C| + |A∩C|) + |A∩B∩C| + Neither. Note that |A∩B| = (only A and B) + |A∩B∩C| = 20 + 15 = 35, |B∩C| = 15 + 15 = 30, and |A∩C| = 10 + 15 = 25. Thus, 200 = 95 + 80 + 75 - (35 + 30 + 25) + 15 + Neither => 200 = 250 - 90 + 15 + Neither => 200 = 175 + Neither => Neither = 25.
Estimated Time:2m 0s
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Topics

Overlapping Sets and Venn Diagrams

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