There are 43
 students in the University Travel Club. The following information and Venn diagram show how many students traveled to Germany, France, and Spain. How many students have not been to Germany, Spain, or France?

18
 members have visited Germany
22
 have been to Spain
19
 have visited France
10
 have been to Germany and France
8
 have only been to Spain
2
 have only been to Germany
5
 have been to only Spain and France
3
 have been to all three countries
Some of the members have not been to any of the three

We can solve this problem by using the principle of **inclusion-exclusion** to calculate how many students have visited any of the three countries (Germany, France, or Spain). Then, we subtract that from the total number of students (43) to find out how many haven't visited any of the countries.

### Given Information:

- Total students: 43
- \( G \) (Germany): 18 students
- \( S \) (Spain): 22 students
- \( F \) (France): 19 students
- \( G \cap F \) (Germany and France): 10 students
- \( G \cap S \cap F \) (Germany, France, and Spain): 3 students
- Only Germany: 2 students
- Only Spain: 8 students
- Only Spain and France: 5 students

### Step 1: Calculate unique groups from the given data.

We know:
- Only Germany: 2 students.
- Only Spain: 8 students.
- Only Spain and France: 5 students.
- All three countries: 3 students.

Next, we need to calculate how many have visited **Germany and France** but not Spain, and **Germany and Spain** but not France.

#### Germany and France (but not Spain):
This is given as \( G \cap F \), but we need to exclude those who visited all three countries.
\[
(G \cap F) \text{ but not Spain} = 10 - 3 = 7.
\]

#### Germany and Spain (but not France):
To find this, we need to subtract those who visited all three countries from the total who visited both Germany and Spain.
However, this information isn't directly given. But using inclusion-exclusion, we can later calculate this missing piece.

### Step 2: Use the principle of inclusion-exclusion.

The formula for finding the number of people who visited at least one of the countries is:
\[
|G \cup F \cup S| = |G| + |F| + |S| - |G \cap F| - |G \cap S| - |F \cap S| + |G \cap F \cap S|.
\]
We already have \( |G| = 18 \), \( |F| = 19 \), and \( |S| = 22 \), and \( G \cap F = 10 \), and \( G \cap F \cap S = 3 \).

To find \( G \cap S \) and \( F \cap S \), we need to figure out how many have been to **Germany and Spain** and **France and Spain**.



There are 43 students in the University Travel Club. The following information and Venn diagram show how many students traveled to Germany, France, and Spain. How many students have not been to Germany, Spain, or France?

18 members have visited Germany
22 have been to Spain
19 have visited France
10 have been to Germany and France
8 have only been to Spain
2 have only been to Germany
5 have been to only Spain and France
3 have been to all three countries
Some of the members have not been to any of the three

This problem involves calculating how many students in a travel club visited Germany, France, and Spain using the inclusion-exclusion principle. The solution shows how to apply set theory to find how many students have not traveled to any of the three countries, based on provided data.

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