A market research firm surveyed a group of 200 consumers regarding their active subscripti… — GRE General Test

A market research firm surveyed a group of $200$ consumers regarding their active subscriptions to three digital services: FilmStream ( $F$ ), AudioVibe ( $A$ ), and PrintPlus ( $P$ ). Every surveyed consumer subscribed to at least one of the three services. The survey gathered the following information:
- $120$ consumers subscribed to FilmStream
- $100$ consumers subscribed to AudioVibe
- $75$ consumers subscribed to PrintPlus
- $45$ consumers subscribed to both FilmStream and AudioVibe
- $35$ consumers subscribed to both AudioVibe and PrintPlus
- $30$ consumers subscribed to both FilmStream and PrintPlus

How many consumers subscribed to all three digital services?

$15$ Answer
B
$25$
C
$35$
D
$55$
E
$95$

Answer

15 consumers subscribed to all three digital services.

According to the 3-set inclusion-exclusion principle,

|F \cup A \cup P| = |F| + |A| + |P| - (|F \cap A| + |A \cap P| + |F \cap P|) + |F \cap A \cap P|

. Substituting the given values gives

200 = 120 + 100 + 75 - (45 + 35 + 30) + |F \cap A \cap P|

, which simplifies to

200 = 185 + |F \cap A \cap P|

. Subtracting

185

from

200

yields

15

Step-by-Step Solution

Apply the Principle of Inclusion-Exclusion for three sets.

|F \cup A \cup P| = |F| + |A| + |P| - (|F \cap A| + |A \cap P| + |F \cap P|) + |F \cap A \cap P|

This formula accounts for elements counted multiple times across overlapping sets.

Substitute the known values into the equation.

200 = 120 + 100 + 75 - (45 + 35 + 30) + |F \cap A \cap P|

Every consumer subscribes to at least one service, so

|F \cup A \cup P| = 200

Simplify the numerical terms on the right side of the equation.

200 = 295 - 110 + |F \cap A \cap P| \implies 200 = 185 + |F \cap A \cap P|

Sum of individual sets is

295

, and sum of pairwise intersections is

110

Solve for the target three-set intersection

|F \cap A \cap P|

|F \cap A \cap P| = 200 - 185 = 15

Isolating the variable gives the number of consumers in all three sets.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Estimated Time:1m 30s

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