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Difficulty: HardOverlapping Sets and Inclusion-Exclusion Principle

A market research firm surveyed 300300 consumers regarding their subscriptions to three streaming services: Service X, Service Y, and Service Z.

- 4040 consumers subscribed to none of the three services.
- 160160 consumers subscribed to Service X.
- 140140 consumers subscribed to Service Y.
- 130130 consumers subscribed to Service Z.
- Exactly 6060 consumers subscribed to exactly two of the services.

How many consumers subscribed to all three services?

  1. A
    35
  2. B
    50
  3. 55Answer
  4. D
    65
  5. E
    110

Answer

55 consumers subscribed to all three services.
The total number of consumers subscribing to at least one service is 300−40=260300 - 40 = 260. Expressing the total union in terms of non-overlapping region counts gives e1+e2+e3=260e_1 + e_2 + e_3 = 260, where e1e_1 is the number subscribing to exactly one service, e2=60e_2 = 60 is the number subscribing to exactly two, and e3e_3 is the number subscribing to all three. Thus, e1+e3=200e_1 + e_3 = 200. Summing the individual totals gives ∣X∣+∣Y∣+∣Z∣=e1+2e2+3e3=430|X| + |Y| + |Z| = e_1 + 2e_2 + 3e_3 = 430, which simplifies to e1+3e3=310e_1 + 3e_3 = 310. Subtracting e1+e3=200e_1 + e_3 = 200 from e1+3e3=310e_1 + 3e_3 = 310 yields 2e3=1102e_3 = 110, so e3=55e_3 = 55.

Step-by-Step Solution

1
Calculate the total number of consumers who subscribed to at least one streaming service.
Total in union ∣X∪Y∪Z∣=300−40=260|X \cup Y \cup Z| = 300 - 40 = 260.
Consumers subscribing to at least one service represent the total sample minus those subscribing to none.
2
Set up the region equations for 3-set inclusion-exclusion in terms of elements in exactly 1 set (e1e_1), exactly 2 sets (e2e_2), and all 3 sets (e3e_3).
Equation 1: e1+e2+e3=260e_1 + e_2 + e_3 = 260; Equation 2: ∣X∣+∣Y∣+∣Z∣=e1+2e2+3e3=160+140+130=430|X| + |Y| + |Z| = e_1 + 2e_2 + 3e_3 = 160 + 140 + 130 = 430.
Summing individual set sizes counts elements in exactly one set once, elements in exactly two sets twice, and elements in all three sets three times.
3
Substitute e2=60e_2 = 60 into both equations.
Equation 1: e1+60+e3=260  ⟹  e1+e3=200e_1 + 60 + e_3 = 260 \implies e_1 + e_3 = 200; Equation 2: e1+2(60)+3e3=430  ⟹  e1+3e3=310e_1 + 2(60) + 3e_3 = 430 \implies e_1 + 3e_3 = 310.
Simplifying both equations yields a system of two linear equations in terms of e1e_1 and e3e_3.
4
Subtract the simplified Equation 1 from Equation 2 to solve for e3e_3.
(e1+3e3)−(e1+e3)=310−200  ⟹  2e3=110  ⟹  e3=55(e_1 + 3e_3) - (e_1 + e_3) = 310 - 200 \implies 2e_3 = 110 \implies e_3 = 55.
Subtracting eliminates e1e_1, directly yielding the number of consumers subscribing to all three services.

Key Concept

3-Set Inclusion-Exclusion Principle using Exact Region Counts
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Topics

Overlapping Sets and Inclusion-Exclusion Principle

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