In a small town, there are three clubs: the Art Club, the Music Club, and the Dance Club. The total number of people in all three clubs combined is 25. However, many people are mem... In a small town, there are three clubs: the Art Club, the Music Club, and the Dance Club. The total number of people in all three clubs combined is 25. However, many people are members of more than one club. Specifically, there are 5 people who are members of both the Art Club and the Music Club, 5 people who are members of both the Music Club and the Dance Club, and 5 people who are members of both the Dance Club and the Art Club. Additionally, there are exactly 3 people who are members of all three clubs. Read more

Steps to Solve

1. Define variables

Let $A$, $M$, and $D$ represent the number of people in the Art Club, Music Club, and Dance Club, respectively.

Let $AM$, $MD$, and $DA$ represent the number of people in both Art and Music Club, Music and Dance Club, and Dance and Art Club, respectively.

Let $AMD$ represent the number of people in all three clubs.

We are given: $AM = 5$ $MD = 5$ $DA = 5$ $AMD = 3$ $A \cup M \cup D = 25$

2. Use the Principle of Inclusion-Exclusion

The total number of people in any of the three clubs is given by: $|A \cup M \cup D| = |A| + |M| + |D| - |A \cap M| - |M \cap D| - |D \cap A| + |A \cap M \cap D|$

We can rewrite this using our defined variables as: $A \cup M \cup D = A + M + D - AM - MD - DA + AMD$

3. Plug in the given values

Substitute the known values into the equation: $25 = A + M + D - 5 - 5 - 5 + 3$ $25 = A + M + D - 15 + 3$ $25 = A + M + D - 12$

4. Solve for $A + M + D$

Add 12 to both sides of the equation: $25 + 12 = A + M + D$ $37 = A + M + D$

Thus, the sum of the number of people in each club is 37. This counts the people in multiple clubs multiple times.

5. Find the number of people who are members of exactly two clubs only

To find the number of members that participate in exactly 2 clubs we need to subtract the members that participate in all three clubs from each pair. $AM_{only} = AM - AMD = 5 - 3 = 2$ $MD_{only} = MD - AMD = 5 - 3 = 2$ $DA_{only} = DA - AMD = 5 - 3 = 2$

So, the number of people who participate in exactly two clubs only is $AM_{only} + MD_{only} + DA_{only} = 2 + 2 + 2 = 6$.

6. Find the number of people who are members of one or more clubs

Let $A_{only}, M_{only}, D_{only}$ be the number of members that participate only in one club. $A \cup M \cup D = A_{only} + M_{only} + D_{only} + AM_{only} + MD_{only} + DA_{only} + AMD$ $25 = A_{only} + M_{only} + D_{only} + 2 + 2 + 2 + 3$ $25 = A_{only} + M_{only} + D_{only} + 9$ $A_{only} + M_{only} + D_{only} = 25 - 9 = 16$

The number of people that participate in all the clubs is $A \cup M \cup D=25$. Number of people that participate in exactly two clubs only is 6. Number of people that participate in all three clubs is 3, number of people that participate only in one club is 16.