In a survey of 150 subscribers of a digital media platform, 75 prefer High-Definition Audi… — JAMB UTME

In a survey of $150$ subscribers of a digital media platform, $75$ prefer High-Definition Audio ( $H$ ), $70$ prefer Offline Downloads ( $D$ ), and $65$ prefer Ad-free Listening ( $A$ ). It is observed that $35$ subscribers prefer both $H$ and $D$ , $30$ prefer both $D$ and $A$ , and $25$ prefer both $H$ and $A$ . If $15$ subscribers prefer none of these three features, how many subscribers prefer exactly two of these features?

Answer: 45

Answer

The number of subscribers who prefer exactly two of the features is 45.

To find the number of subscribers who prefer exactly two features, we first calculate the cardinality of the union of all three sets as

150 - 15 = 135

. Applying the 3-set inclusion-exclusion formula gives the number of subscribers preferring all three features as

15

. Subtracting

15

from each pairwise intersection gives the exclusive regions:

20

for

H

and

D

only,

15

for

D

and

A

only, and

10

for

H

and

A

only. Summing these three exclusive regions gives

20 + 15 + 10 = 45

Step-by-Step Solution

Determine the total number of subscribers who prefer at least one feature.

|H \cup D \cup A| = 135

Subtracting the number of subscribers who prefer none of the features (

15

) from the universal set size (

150

) gives

|H \cup D \cup A| = 150 - 15 = 135

Apply the Principle of Inclusion-Exclusion for three sets to find the number of subscribers who prefer all three features.

|H \cap D \cap A| = 15

Substitute the known cardinalities into

|H \cup D \cup A| = |H| + |D| + |A| - |H \cap D| - |D \cap A| - |H \cap A| + |H \cap D \cap A|

to get

135 = 75 + 70 + 65 - (35 + 30 + 25) + |H \cap D \cap A|

, which simplifies to

135 = 120 + |H \cap D \cap A|

, giving

|H \cap D \cap A| = 15

Calculate the number of subscribers preferring exactly two features by subtracting the triple intersection from each pairwise intersection.

45 subscribers

Subscribers preferring only

H

and

D = 35 - 15 = 20

, only

D

and

A = 30 - 15 = 15

, and only

H

and

A = 25 - 15 = 10

. Summing these exclusive regions yields

20 + 15 + 10 = 45

Key Concept

Cardinality of set operations and 3-set inclusion-exclusion principle

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