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Difficulty: Very hardOverlapping Sets and Venn Diagrams

At an intellectual property law firm, a senior partner reviewed 180180 patent applications across three technical domains: Artificial Intelligence, Biotechnology, and Clean Energy. The audit revealed that 2525 applications belonged to none of these three domains. Furthermore, 9090 applications involved Artificial Intelligence, 8585 involved Biotechnology, and 8080 involved Clean Energy. If the number of applications that belonged to exactly two of these domains was three times the number of applications that belonged to all three domains, how many patent applications belonged to exactly one domain?

Answer: 75 applications

Answer

The total number of patent applications that belonged to exactly one domain is 7575.
The total number of applications in at least one domain is 180−25=155180 - 25 = 155. Let E1E_1 be the number of applications in exactly one domain, E2E_2 in exactly two domains, and xx in all three domains. The total union is E1+E2+x=155E_1 + E_2 + x = 155, while the sum of individual set sizes is ∣A∣+∣B∣+∣C∣=E1+2E2+3x=90+85+80=255|A| + |B| + |C| = E_1 + 2E_2 + 3x = 90 + 85 + 80 = 255. Subtracting the union equation from the sum equation gives E2+2x=100E_2 + 2x = 100. Given E2=3xE_2 = 3x, substituting gives 5x=100  ⟹  x=205x = 100 \implies x = 20, which implies E2=60E_2 = 60. Finally, subtracting E2E_2 and xx from the total union yields E1=155−60−20=75E_1 = 155 - 60 - 20 = 75.

Step-by-Step Solution

1
Calculate the total number of applications in at least one domain
∣A∪B∪C∣=180−25=155|A \cup B \cup C| = 180 - 25 = 155
Subtracting the applications belonging to none of the domains from the overall total yields the total number of unique applications covered by the three domains combined.
2
Formulate regional Venn diagram equations
E1+E2+x=155E_1 + E_2 + x = 155 and E1+2E2+3x=255E_1 + 2E_2 + 3x = 255
Summing individual set counts counts elements in exactly two domains twice and elements in all three domains three times.
3
Deduce the relationship between E2E_2 and xx
E2+2x=100E_2 + 2x = 100
Subtracting the equation for total union from the sum of individual sets isolates the overcounted regions.
4
Use the given proportion E2=3xE_2 = 3x to solve for xx and E2E_2
x=20x = 20 and E2=60E_2 = 60
Substituting E2=3xE_2 = 3x into E2+2x=100E_2 + 2x = 100 gives 5x=1005x = 100, so x=20x = 20 and E2=60E_2 = 60.
5
Find the number of applications belonging to exactly one domain (E1E_1)
E1=75E_1 = 75
Subtracting the count of applications in exactly two domains (6060) and all three domains (2020) from the total union (155155) gives 155−60−20=75155 - 60 - 20 = 75.

Key Concept

Three-Set Venn Diagram Region Partitioning
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Topics

Overlapping Sets and Venn Diagrams

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