give an example of word problem solved using venn diagram where the intersection of three sets is unknown

Here's an example of a word problem that can be solved using a Venn diagram, where the intersection of three sets is unknown:

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**Problem:**

A school conducted a survey among 100 students regarding their participation in three extracurricular activities: football, basketball, and chess. The results of the survey are as follows:

- 35 students play football.
- 30 students play basketball.
- 25 students play chess.
- 10 students play both football and basketball.
- 8 students play both basketball and chess.
- 5 students play both football and chess.
- It is also known that 3 students participate in all three activities: football, basketball, and chess.

**Question:** How many students do not participate in any of the three activities?

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### Step-by-Step Solution:

Let:
- \( F \) represent the set of students who play football.
- \( B \) represent the set of students who play basketball.
- \( C \) represent the set of students who play chess.

We are given:
- \( |F| = 35 \) (students who play football),
- \( |B| = 30 \) (students who play basketball),
- \( |C| = 25 \) (students who play chess),
- \( |F \cap B| = 10 \) (students who play both football and basketball),
- \( |B \cap C| = 8 \) (students who play both basketball and chess),
- \( |F \cap C| = 5 \) (students who play both football and chess),
- \( |F \cap B \cap C| = 3 \) (students who play all three activities).

#### 1. Find the number of students who play only two sports:

To find the number of students who play only two sports but not the third, we subtract the number of students who play all three sports (\( |F \cap B \cap C| = 3 \)) from the number of students who play any two sports:
- \( |F \cap B \text{ only}| = 10 - 3 = 7 \) (students who play only football and basketball),
- \( |B \cap C \text{ only}| = 8 - 3 = 5 \) (students who play only basketball and chess),
- \( |F \cap C \text{ only}| = 5 - 3 = 2 \) (students who play only football and chess).

#### 2. Find the number of students who play only one sport:

To find the number of students who play only one sport, we subtract those who play two or three sports from the total number who play each sport:
- \( |F \text{ only}| = 35 - (7 + 2 + 3) = 23 \),
- \( |B \text{ only}| = 30 - (7 + 5 + 3) = 15 \),
- \( |C \text{ only}| = 25 - (2 + 5 + 3) = 15 \).

#### 3. Total number of students participating in at least one activity:

Now, we sum the number of students who participate in exactly one, exactly two, and exactly three activities:
- \( 23 \) (only football) + \( 15 \) (only basketball) + \( 15 \) (only chess) + \( 7 \) (football and basketball only) + \( 5 \) (basketball and chess only) + \( 2 \) (football and chess only) + \( 3 \) (all three activities) = 70.

#### 4. Number of students who do not participate in any activity:

Since there are 100 students in total, the number of students who do not participate in any of the activities is:
- \( 100 - 70 = 30 \).

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### Final Answer:

**30 students do not participate in any of the three activities.**

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Do you want any details or have any questions?

### 5 Related Questions:
1. How would the solution change if 40 students played football instead of 35?
2. What if 4 students participated in all three activities instead of 3?
3. How would you represent this data using a set equation without a Venn diagram?
4. How can this problem be generalized for four different activities?
5. What if you knew the total number of students involved in exactly two activities but not each individual overlap?

**Tip:** Venn diagrams are very effective for visualizing complex relationships between multiple sets and can simplify the process of solving word problems involving groups or categories.

This example demonstrates how to solve a Venn diagram problem involving three sets (football, basketball, chess) and an unknown intersection using set theory and combinatorics. With detailed step-by-step calculations, it's ideal for high school students learning about the Inclusion-Exclusion Principle.

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