When the members of a family discussed where their annual reunion should take​ place, they found that out of all the family​ members, 10 would not go to a​ park, 8 would not go to a​ beach, 12 would not go to the family​ cottage, 6 would go to neither a park nor a​ beach, 6 would go to neither a beach nor the family​ cottage, 8 would go to neither a park nor the family​ cottage, 5 would not go to a park or a beach or to the family​ cottage, and 2 would go to all three places. What is the total number of family​ members?

We are given a problem involving the principle of inclusion and exclusion, where three different events are involved:

- 10 family members would not go to a park (denote this as \( |P^c| = 10 \)).
- 8 family members would not go to a beach (denote this as \( |B^c| = 8 \)).
- 12 family members would not go to the family cottage (denote this as \( |C^c| = 12 \)).
- 6 family members would go to neither a park nor a beach (denote this as \( |P^c \cap B^c| = 6 \)).
- 6 family members would go to neither a beach nor the family cottage (denote this as \( |B^c \cap C^c| = 6 \)).
- 8 family members would go to neither a park nor the family cottage (denote this as \( |P^c \cap C^c| = 8 \)).
- 5 family members would not go to a park, a beach, or the family cottage (denote this as \( |P^c \cap B^c \cap C^c| = 5 \)).
- 2 family members would go to all three places (denote this as \( |P \cap B \cap C| = 2 \)).

We are asked to find the total number of family members \( N \). Using the principle of inclusion and exclusion for three sets (park \( P \), beach \( B \), and cottage \( C \)), we have:

\[
N = |P^c| + |B^c| + |C^c| - |P^c \cap B^c| - |B^c \cap C^c| - |P^c \cap C^c| + |P^c \cap B^c \cap C^c|
\]

Let me calculate this.The total number of family members is 15.

Would you like a detailed explanation of how the principle of inclusion and exclusion was applied, or have any questions?

Here are some related questions for further exploration:
1. What if 3 additional members could go to all three places? How would the total change?
2. How does the principle of inclusion-exclusion work for four sets?
3. Can the formula be used to determine how many members can only attend one specific location?
4. What if we knew how many family members went only to the park? How would that affect the total?
5. How would the result change if no one went to the family cottage?

**Tip:** The inclusion-exclusion principle helps in counting elements that belong to multiple sets by avoiding over-counting those in intersections.

When the members of a family discussed where their annual reunion should take place, they found that out of all the family members, 10 would not go to a park, 8 would not go to a beach, 12 would not go to the family cottage, 6 would go to neither a park nor a beach, 6 would go to neither a beach nor the family cottage, 8 would go to neither a park nor the family cottage, 5 would not go to a park or a beach or to the family cottage, and 2 would go to all three places. What is the total number of family members?

Learn how to solve the family reunion problem using the principle of inclusion and exclusion. This problem involves set theory concepts and covers calculating intersections and exclusions of multiple sets. Suitable for students in Grades 9-12.

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