A survey of 600
 students yielded the following information: 126
 were seniors, 248
 were commuters, and 66
 were seniors and commuters. How many students were neither seniors nor commuters?

We can solve this problem using the principle of set theory, specifically the inclusion-exclusion principle. Here's how:

Let:
- \( S \) represent the set of seniors.
- \( C \) represent the set of commuters.

We are given the following dataWe can solve this problem using the principle of set theory, specifically the inclusion-exclusion principle. Here's how:

Let:
- \( S \) represent the set of seniors.
- \( C \) represent the set of commuters.

We are given the following 
- Total students = 600
- Number of seniors (\( |S| \)) = 126
- Number of commuters (\( |C| \)) = 248
- Number of seniors who are also commuters (\( |S \cap C| \)) = 66

We need to find how many students were neither seniors nor commuters. This can be calculated by first finding how many students were either seniors or commuters, and then subtracting this number from the total number of students.

### Step 1: Use the inclusion-exclusion principle
The inclusion-exclusion principle for two sets is:

\[
|S \cup C| = |S| + |C| - |S \cap C|
\]

Substituting the values:

\[
|S \cup C| = 126 + 248 - 66 = 308
\]

This means that 308 students are either seniors, commuters, or both.

### Step 2: Find students who are neither seniors nor commuters
The number of students who are neither seniors nor commuters is:

\[
600 - |S \cup C| = 600 - 308 = 292
\]

### Conclusion:
292 students are neither seniors nor commuters.

Would you like more details, or do you have any questions?

Here are 5 questions related to this problem:
1. How does the inclusion-exclusion principle apply to more than two sets?
2. What would the result be if 30 students were added to the total and they were all commuters but not seniors?
3. Can you derive the formula for the inclusion-exclusion principle?
4. What happens if there is no overlap between seniors and commuters?
5. How does set theory relate to probability in surveys?

**Tip**: Always double-check if data overlaps when applying set theory to real-world problems!

A survey of 600 students yielded the following information: 126 were seniors, 248 were commuters, and 66 were seniors and commuters. How many students were neither seniors nor commuters?

Solve a survey problem involving 600 students, where 126 are seniors, 248 are commuters, and 66 are both. Learn how to apply the inclusion-exclusion principle to find how many students are neither seniors nor commuters. Suitable for Grades 9-11.

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