A market research study examined the streaming service subscriptions of 250 households. Ev… — GMAT

A market research study examined the streaming service subscriptions of $250$ households. Every surveyed household subscribes to at least one of three streaming services: Service X, Service Y, or Service Z.

- $130$ households subscribe to Service X.
- $110$ households subscribe to Service Y.
- $120$ households subscribe to Service Z.
- $40$ households subscribe to both Service X and Service Y.
- $35$ households subscribe to both Service Y and Service Z.
- $45$ households subscribe to both Service X and Service Z.

How many households subscribe to all three streaming services?

Answer: 10 households

Answer

10 households subscribe to all three streaming services.

According to the Principle of Inclusion-Exclusion for three sets,

|X \cup Y \cup Z| = |X| + |Y| + |Z| - (|X \cap Y| + |Y \cap Z| + |X \cap Z|) + |X \cap Y \cap Z|

. Since every household subscribes to at least one service,

|X \cup Y \cup Z| = 250

. Substituting the values yields

250 = 130 + 110 + 120 - (40 + 35 + 45) + |X \cap Y \cap Z|

, which simplifies to

250 = 240 + |X \cap Y \cap Z|

. Solving for

|X \cap Y \cap Z|

gives

10

Step-by-Step Solution

Set up the Inclusion-Exclusion formula for three overlapping sets

Total = |X| + |Y| + |Z| - (|X ∩ Y| + |Y ∩ Z| + |X ∩ Z|) + |X ∩ Y ∩ Z|

Because every household subscribes to at least one service, the total number of households equals the union of all three sets.

Sum the individual set sizes and the pairwise intersection sizes

Sum of individual sets = 130 + 110 + 120 = 360; Sum of pairwise intersections = 40 + 35 + 45 = 120

Aggregating individual and overlapping counts simplifies substitution.

Substitute the values into the formula and solve for the intersection of all three sets

250 = 360 - 120 + |X ∩ Y ∩ Z| => 250 = 240 + |X ∩ Y ∩ Z| => |X ∩ Y ∩ Z| = 10

Subtracting 240 from 250 yields the exact count of households subscribing to all three services.

Key Concept

Inclusion-Exclusion Principle for Three Sets

Estimated Time:1m 30s

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