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Difficulty: HardSets and Set Operations

In a survey of 160 music enthusiasts, 80 listen to Afrobeat, 70 listen to Highlife, and 65 listen to Reggae. It was found that 30 listen to both Afrobeat and Highlife, 25 listen to both Highlife and Reggae, and 28 listen to both Afrobeat and Reggae. If 15 enthusiasts listen to none of these three genres, how many enthusiasts listen to exactly two of these genres?

Answer: 44

Answer

The number of enthusiasts who listen to exactly two of the three genres is 44.
The correct answer is 44. By subtracting the 15 non-listeners from 160, the union of the three sets contains 145 members. Solving the 3-set inclusion-exclusion equation yields 13 enthusiasts who listen to all three genres. Subtracting 13 from each given pairwise intersection gives 17, 12, and 15 for the regions representing listeners of exactly two genres. Summing these three counts yields 44.

Step-by-Step Solution

1
Calculate the cardinality of the union of all three sets.
n(A∪H∪R)=160−15=145n(A \cup H \cup R) = 160 - 15 = 145
Subtracting the number of enthusiasts listening to none of the genres from the universal set.
2
Apply the Principle of Inclusion-Exclusion for 3 sets to find the triple intersection.
n(A∩H∩R)=13n(A \cap H \cap R) = 13
Substituting known cardinalities into n(A∪H∪R)=n(A)+n(H)+n(R)−[n(A∩H)+n(H∩R)+n(A∩R)]+n(A∩H∩R)n(A \cup H \cup R) = n(A)+n(H)+n(R) - [n(A \cap H)+n(H \cap R)+n(A \cap R)] + n(A \cap H \cap R).
3
Calculate the count of enthusiasts in each 'exactly two genres' region.
Afrobeat and Highlife only = 17; Highlife and Reggae only = 12; Afrobeat and Reggae only = 15.
Subtracting the triple intersection count (1313) from each pairwise intersection.
4
Sum the counts of the three distinct two-genre regions.
17+12+15=4417 + 12 + 15 = 44
Combining all mutually exclusive regions representing enthusiasts of exactly two genres.

Key Concept

Principle of Inclusion-Exclusion for three sets and Venn diagram region partitioning.
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Topics

Sets and Set Operations

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