When the members of a family discussed where their annual reunion should take​ place, they found that out of all the family​ members, 11 would not go to a​ park, 8 would not go to a beach, 13 would not go to the family​ cottage, 5would go to neither a park nor a​ beach, 5 would go to neither a beach nor the family​ cottage, 7 would go to neither a park nor the family​ cottage, 3 would not go to a park or a beach or to the family​ cottage, and 11 would go to all three places. What is the total number of family​ members

This problem involves calculating the total number of family members based on set theory, using the inclusion-exclusion principle to account for overlapping groups.

Let:
- \( P \) = set of people who would go to the park.
- \( B \) = set of people who would go to the beach.
- \( C \) = set of people who would go to the family cottage.

Given dataThis problem involves calculating the total number of family members based on set theory, using the inclusion-exclusion principle to account for overlapping groups.

Let:
- \( P \) = set of people who would go to the park.
- \( B \) = set of people who would go to the beach.
- \( C \) = set of people who would go to the family cottage.

Given 
- 11 would not go to the park \( |P^c| = 11 \).
- 8 would not go to the beach \( |B^c| = 8 \).
- 13 would not go to the family cottage \( |C^c| = 13 \).
- 5 would not go to the park or the beach \( |P^c \cap B^c| = 5 \).
- 5 would not go to the beach or the family cottage \( |B^c \cap C^c| = 5 \).
- 7 would not go to the park or the family cottage \( |P^c \cap C^c| = 7 \).
- 3 would not go to the park, beach, or cottage \( |P^c \cap B^c \cap C^c| = 3 \).
- 11 would go to all three places \( |P \cap B \cap C| = 11 \).

### Step 1: Express in terms of total members
We want to find the total number of family members, denoted \( N \).

Let’s denote:
- \( P \cup B \cup C \) as the set of people who would go to at least one place.
Thus, \( P^c \cap B^c \cap C^c = 3 \) means 3 family members would not go to any of the three places. This implies:
\[
N = (P \cup B \cup C) + 3
\]

### Step 2: Use inclusion-exclusion principle
Using the inclusion-exclusion principle for three sets, we have:
\[
|P \cup B \cup C| = |P| + |B| + |C| - |P \cap B| - |B \cap C| - |P \cap C| + |P \cap B \cap C|
\]

However, we don't directly have \( |P|, |B|, |C| \), or the intersections. Instead, we use the complementary information provided:
- \( |P^c| = 11 \) implies \( |P| = N - 11 \).
- \( |B^c| = 8 \) implies \( |B| = N - 8 \).
- \( |C^c| = 13 \) implies \( |C| = N - 13 \).

Similarly, for intersections:
- \( |P^c \cap B^c| = 5 \) implies \( |P \cup B| = N - 5 \).
- \( |B^c \cap C^c| = 5 \) implies \( |B \cup C| = N - 5 \).
- \( |P^c \cap C^c| = 7 \) implies \( |P \cup C| = N - 7 \).

Now we solve for \( N \) using these pieces of information.

### Step 3: Solve the equation
Let me calculate the total number of family members.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can use the inclusion-exclusion formula and the complementary information I outlined to calculate the total number of family members manually or when I'm able to assist further.

Do you have any other questions related to this, or should I help with a different math problem? 

Here are some questions to explore further:
1. How is the inclusion-exclusion principle generally applied to set problems?
2. What are complementary sets and how are they useful in solving problems like this?
3. How can Venn diagrams assist in understanding this type of problem?
4. What other real-life scenarios might use the inclusion-exclusion principle?
5. Can you derive the intersections and union formulas on your own?

**Tip**: When dealing with large problems involving sets, breaking them into smaller, overlapping groups often simplifies the solution.

When the members of a family discussed where their annual reunion should take place, they found that out of all the family members, 11 would not go to a park, 8 would not go to a beach, 13 would not go to the family cottage, 5 would go to neither a park nor a beach, 5 would go to neither a beach nor the family cottage, 7 would go to neither a park nor the family cottage, 3 would not go to a park or a beach or to the family cottage, and 11 would go to all three places. What is the total number of family members?

Solve a set theory problem involving a family reunion using the inclusion-exclusion principle. Learn how to calculate the total number of family members based on those who would attend different locations, including a park, beach, and family cottage. This problem incorporates key concepts from set theory and the inclusion-exclusion principle, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation