A university surveyed a cohort of 150 freshmen regarding their membership in three student… — GRE General Test

A university surveyed a cohort of $150$ freshmen regarding their membership in three student organizations: the Art Club ( $A$ ), the Music Society ( $M$ ), and the Theater Guild ( $T$ ). The survey revealed the following data:

- $68$ students belong to the Art Club.
- $62$ students belong to the Music Society.
- $54$ students belong to the Theater Guild.
- $22$ students belong to both the Art Club and the Music Society.
- $18$ students belong to both the Music Society and the Theater Guild.
- $15$ students belong to both the Art Club and the Theater Guild.
- $8$ students belong to all three organizations.

How many of the surveyed students belong to exactly one of these three organizations?

Answer: 98 students

Answer

98 students belong to exactly one of the three organizations.

To find the number of students belonging to exactly one organization, analyze the regions of a 3-set Venn diagram starting from the innermost region (all three clubs =

8

). Subtracting

8

from each pairwise intersection gives the students in exactly two clubs: Art & Music only (

14

), Music & Theater only (

10

), and Art & Theater only (

7

). Next, subtract the overlapping regions from each club total: Art only is

68 - (14 + 7 + 8) = 39

; Music only is

62 - (14 + 10 + 8) = 30

; Theater only is

54 - (7 + 10 + 8) = 29

. Summing these single-club regions gives

39 + 30 + 29 = 98

Step-by-Step Solution

Find the number of students belonging strictly to each pair of organizations (two-set intersections only).

Art and Music only =

14

; Music and Theater only =

10

; Art and Theater only =

7

The given pairwise totals include students who belong to all three organizations (

8

), so subtracting

8

isolates those in exactly two groups.

Determine the number of students belonging to each individual organization exclusively.

Art only =

39

; Music only =

30

; Theater only =

29

Subtract all shared membership regions (both two-group only and three-group) from each total organization membership.

Add the counts of students belonging to exactly one group.

39 + 30 + 29 = 98

The question requests the sum of all students in the non-overlapping single-set regions.

Key Concept

Three-Set Venn Diagram Region Partitioning

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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