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Difficulty: HardSet Theory Concepts and Venn Diagrams

A survey of 200200 financial executives analyzed their personal investment portfolios across three asset classes: Equities (EE), Real Estate (RR), and Commodities (CC). Exactly 2020 executives hold no investments in any of these three asset classes. Among the remaining executives, 115115 hold Equities, 9595 hold Real Estate, and 8080 hold Commodities. Furthermore, 4545 executives hold both Equities and Real Estate, 3535 hold both Real Estate and Commodities, and 4040 hold both Equities and Commodities. How many of the surveyed executives hold investments in exactly two of these three asset classes?

  1. A
    80
  2. 90Answer
  3. C
    100
  4. D
    110
  5. E
    120

Answer

90
To find the number of executives holding investments in exactly two asset classes, we first determine the total number of executives invested in at least one asset class: 200−20=180200 - 20 = 180. Applying the three-set inclusion-exclusion formula, 180=115+95+80−(45+35+40)+∣E∩R∩C∣180 = 115 + 95 + 80 - (45 + 35 + 40) + |E \cap R \cap C|, which simplifies to 180=170+∣E∩R∩C∣180 = 170 + |E \cap R \cap C|, giving ∣E∩R∩C∣=10|E \cap R \cap C| = 10. The number of executives holding exactly two asset classes is calculated by subtracting the triple intersection (10) from each pairwise overlap: (45−10)+(35−10)+(40−10)=35+25+30=90(45 - 10) + (35 - 10) + (40 - 10) = 35 + 25 + 30 = 90.

Step-by-Step Solution

1
Calculate the total number of executives holding at least one asset class.
The total number in the union ∣E∪R∪C∣=200−20=180|E \cup R \cup C| = 200 - 20 = 180.
Subtracting executives holding no investments gives the cardinality of the union.
2
Use the Principle of Inclusion-Exclusion for three sets to find the triple intersection ∣E∩R∩C∣|E \cap R \cap C|.
∣E∪R∪C∣=∣E∣+∣R∣+∣C∣−(∣E∩R∣+∣R∩C∣+∣E∩C∣)+∣E∩R∩C∣|E \cup R \cup C| = |E| + |R| + |C| - (|E \cap R| + |R \cap C| + |E \cap C|) + |E \cap R \cap C|, so 180=115+95+80−(45+35+40)+∣E∩R∩C∣180 = 115 + 95 + 80 - (45 + 35 + 40) + |E \cap R \cap C|, yielding 180=290−120+∣E∩R∩C∣=170+∣E∩R∩C∣180 = 290 - 120 + |E \cap R \cap C| = 170 + |E \cap R \cap C|, hence ∣E∩R∩C∣=10|E \cap R \cap C| = 10.
The inclusion-exclusion formula relates the union, individual set sizes, pairwise overlaps, and the triple overlap.
3
Determine the number of executives in each pairwise intersection that belong ONLY to those two sets.
Equities and Real Estate only: 45−10=3545 - 10 = 35; Real Estate and Commodities only: 35−10=2535 - 10 = 25; Equities and Commodities only: 40−10=3040 - 10 = 30.
Subtracting the triple intersection count from each pairwise intersection isolates those holding exactly two asset classes.
4
Sum the counts for executives holding exactly two asset classes.
35+25+30=9035 + 25 + 30 = 90.
Adding these disjoint sets yields the total number of executives in exactly two categories.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Decomposition
Estimated Time:2m 0s
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Topics

Set Theory Concepts and Venn Diagrams

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