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Difficulty: MediumOverlapping Sets and Venn Diagrams

A pharmaceutical research laboratory tested 120120 chemical compounds for the presence of three distinct bio-reactivity markers: Enzyme Inhibition (EE), Protein Binding (PP), and Receptor Affinity (RR). The test results revealed the following:

- 5858 compounds exhibited Enzyme Inhibition (EE).
- 5252 compounds exhibited Protein Binding (PP).
- 4646 compounds exhibited Receptor Affinity (RR).
- 88 compounds exhibited all three markers.
- 1515 compounds exhibited none of the three markers.

How many of the tested compounds exhibited exactly two of the three markers?

  1. A
    2727
  2. 3535Answer
  3. C
    4343
  4. D
    5151
  5. E
    6666

Answer

The number of compounds exhibiting exactly two markers is 3535.
The total number of compounds displaying at least one marker is 120−15=105120 - 15 = 105. The sum of the single set counts is 58+52+46=15658 + 52 + 46 = 156. Using the three-set relation N(A)+N(B)+N(C)−N(union)=N(exactly 2)+2N(all 3)N(A) + N(B) + N(C) - N(\text{union}) = N(\text{exactly 2}) + 2N(\text{all 3}), we get 156−105=N(exactly 2)+2(8)156 - 105 = N(\text{exactly 2}) + 2(8). Simplifying gives 51=N(exactly 2)+1651 = N(\text{exactly 2}) + 16, which yields N(exactly 2)=35N(\text{exactly 2}) = 35.

Step-by-Step Solution

1
Determine the total number of compounds exhibiting at least one marker.
At least one marker=120−15=105\text{At least one marker} = 120 - 15 = 105.
Subtracting the compounds with no markers from the total gives the union size N(E∪P∪R)N(E \cup P \cup R).
2
Calculate the sum of the individual set counts.
N(E)+N(P)+N(R)=58+52+46=156N(E) + N(P) + N(R) = 58 + 52 + 46 = 156.
This sum counts elements with exactly one marker once, elements with exactly two markers twice, and elements with all three markers three times.
3
Apply the 3-set region formula connecting single counts, region overlaps, and total union.
156=(Exactly 1)+2(Exactly 2)+3(All 3)156 = (\text{Exactly 1}) + 2(\text{Exactly 2}) + 3(\text{All 3}) and 105=(Exactly 1)+(Exactly 2)+(All 3)105 = (\text{Exactly 1}) + (\text{Exactly 2}) + (\text{All 3}).
Subtracting the second equation from the first yields: 156−105=(Exactly 2)+2(All 3)156 - 105 = (\text{Exactly 2}) + 2(\text{All 3}).
4
Substitute the known value for all three markers to solve for compounds with exactly two markers.
51=(Exactly 2)+2(8)  ⟹  51=(Exactly 2)+16  ⟹  Exactly 2=3551 = (\text{Exactly 2}) + 2(8) \implies 51 = (\text{Exactly 2}) + 16 \implies \text{Exactly 2} = 35.
Solving the linear algebraic equation yields the exact number of compounds in the pairwise-only regions.

Key Concept

Three-Set Overlapping Venn Diagrams and Region Counting Equations
Estimated Time:2m 0s
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Topics

Overlapping Sets and Venn Diagrams

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