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Difficulty: MediumSet Theory Concepts and Venn Diagrams

A community health center conducted a study of 250250 adults regarding their participation in three wellness programs: Nutrition Counseling (NN), Exercise Coaching (EE), and Stress Management (SS). The survey revealed the following data:

• 125125 adults participate in Nutrition Counseling.
• 105105 adults participate in Exercise Coaching.
• 8585 adults participate in Stress Management.
• 1515 adults participate in all three programs.
• 3030 adults participate in none of the three programs.

How many adults participate in exactly two of the three programs?

Answer: 65 adults

Answer

65
The total number of surveyed adults is 250250, and 3030 participate in none of the programs, meaning 220220 adults participate in at least one program. Let x1x_1 be the number of adults in exactly one program, x2x_2 in exactly two programs, and x3=15x_3 = 15 in all three programs. We have x1+x2+15=220x_1 + x_2 + 15 = 220, which simplifies to x1+x2=205x_1 + x_2 = 205. Additionally, summing the individual program participants yields ∣N∣+∣E∣+∣S∣=125+105+85=315|N| + |E| + |S| = 125 + 105 + 85 = 315. By region expansion, ∣N∣+∣E∣+∣S∣=x1+2x2+3x3|N| + |E| + |S| = x_1 + 2x_2 + 3x_3. Substituting x3=15x_3 = 15 gives x1+2x2+45=315x_1 + 2x_2 + 45 = 315, or x1+2x2=270x_1 + 2x_2 = 270. Subtracting x1+x2=205x_1 + x_2 = 205 from x1+2x2=270x_1 + 2x_2 = 270 gives x2=65x_2 = 65.

Step-by-Step Solution

1
Determine the number of adults in the union of all three set categories
∣N∪E∪S∣=250−30=220|N \cup E \cup S| = 250 - 30 = 220
Subtracting the individuals participating in none of the programs from the total surveyed yields the total count of individuals participating in at least one program.
2
Set up an equation for the total unique participants using disjoint region variables
x1+x2=205x_1 + x_2 = 205
The union equals x1+x2+x3=220x_1 + x_2 + x_3 = 220, where x1x_1 represents adults in exactly 1 program, x2x_2 in exactly 2, and x3=15x_3 = 15 in all 3 programs.
3
Set up an equation using the sum of the individual program totals
x1+2x2=270x_1 + 2x_2 = 270
The sum ∣N∣+∣E∣+∣S∣=125+105+85=315|N| + |E| + |S| = 125 + 105 + 85 = 315 counts single-program participants once, double-program participants twice, and triple-program participants three times (x1+2x2+3(15)=315x_1 + 2x_2 + 3(15) = 315).
4
Solve the system of linear equations for x2x_2
x2=270−205=65x_2 = 270 - 205 = 65
Subtracting (x1+x2=205)(x_1 + x_2 = 205) from (x1+2x2=270)(x_1 + 2x_2 = 270) isolates x2x_2, which is the exact number of adults participating in exactly two programs.

Key Concept

3-Set Inclusion-Exclusion Principle & Venn Diagram Region Partitioning
Estimated Time:1m 45s
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Topics

Set Theory Concepts and Venn Diagrams

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