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Zorluk: Çok zorOverlapping Sets and Inclusion-Exclusion Principle

At a financial analytics firm, a group of 200 analysts were evaluated on their proficiency in three software tools: Options Analytics, Futures Trader, and Swaps Pricing. Exactly 15% of the analysts had no proficiency in any of the three tools. Among the remaining analysts, 110 were proficient in Options Analytics, 95 were proficient in Futures Trader, and 85 were proficient in Swaps Pricing. If exactly 20 analysts were proficient in all three tools, how many analysts were proficient in exactly one of the three tools?

  1. A
    10
  2. B
    30
  3. C
    50
  4. 70Cevap
  5. E
    100

Cevap

70 analysts were proficient in exactly one of the three tools.
The total number of analysts is 200, and 15% (30 analysts) have no proficiency, leaving 170 analysts proficient in at least one tool. Applying the 3-set inclusion-exclusion formula N(A∪B∪C)=N(A)+N(B)+N(C)−S2+N(A∩B∩C)N(A \cup B \cup C) = N(A) + N(B) + N(C) - S_2 + N(A \cap B \cap C), we get 170=110+95+85−S2+20170 = 110 + 95 + 85 - S_2 + 20, which yields S2=140S_2 = 140. Since S2S_2 counts elements in exactly two sets once and elements in all three sets three times, the number of analysts proficient in exactly two tools is 140−3(20)=80140 - 3(20) = 80. Finally, subtracting those proficient in exactly two tools (80) and all three tools (20) from the total proficient in at least one tool (170) gives 170−80−20=70170 - 80 - 20 = 70.

Adım Adım Çözüm

1
Calculate the total number of analysts proficient in at least one tool
At least one=200−(0.15×200)=200−30=170\text{At least one} = 200 - (0.15 \times 200) = 200 - 30 = 170
Analysts who are not proficient in any tool must be excluded from the total group size to find the union of the three sets.
2
Apply the 3-set inclusion-exclusion formula to find the sum of pairwise intersections
170=110+95+85−S2+20  ⟹  170=310−S2  ⟹  S2=140170 = 110 + 95 + 85 - S_2 + 20 \implies 170 = 310 - S_2 \implies S_2 = 140, where S2=N(Options∩Futures)+N(Futures∩Swaps)+N(Options∩Swaps)S_2 = N(\text{Options} \cap \text{Futures}) + N(\text{Futures} \cap \text{Swaps}) + N(\text{Options} \cap \text{Swaps})
The standard inclusion-exclusion principle states that N(A∪B∪C)=N(A)+N(B)+N(C)−S2+N(A∩B∩C)N(A \cup B \cup C) = N(A) + N(B) + N(C) - S_2 + N(A \cap B \cap C).
3
Determine the number of analysts proficient in exactly two tools
Exactly 2=S2−3×N(All 3)=140−3(20)=140−60=80\text{Exactly 2} = S_2 - 3 \times N(\text{All 3}) = 140 - 3(20) = 140 - 60 = 80
Each member of the triple intersection is counted 3 times in S2S_2. Subtracting 3×N(All 3)3 \times N(\text{All 3}) isolates the elements belonging to exactly two sets.
4
Calculate the number of analysts proficient in exactly one tool
Exactly 1=N(At least 1)−Exactly 2−N(All 3)=170−80−20=70\text{Exactly 1} = N(\text{At least 1}) - \text{Exactly 2} - N(\text{All 3}) = 170 - 80 - 20 = 70
The union of the three sets consists of elements proficient in exactly 1 tool, exactly 2 tools, and all 3 tools.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Subset Decomposition
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Konular

Overlapping Sets and Inclusion-Exclusion Principle

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