At a community​ college, a survey was taken to determine where students study on campus. Of the 250 students​ surveyed, it was determined that 171 studied in the library. 131 studied in the cafeteria. 79 studied in both the library and the cafeteria. Complete parts ​a) through ​c). A)Of those students​ surveyed, how many studied only in the​ library? B)Of those students​ surveyed, how many studied only in the​ cafeteria? C)Of those students​ surveyed, how many did not study in either of these​ places?

To solve the problem, we can use the principle of inclusion-exclusion, which helps us determine the number of elements in different subsets.

### Given:
- Total number of students surveyed: \( n = 250 \)
- Number of students who studied in the library: \( |L| = 171 \)
- Number of students who studied in the cafeteria: \( |C| = 131 \)
- Number of students who studied in both the library and the cafeteria: \( |L \cap C| = 79 \)

### Part A: Number of students who studied only in the library
Students who studied only in the library can be calculated by subtracting those who studied in both places from those who studied in the library.

\[
\text{Students who studied only in the library} = |L| - |L \cap C|
\]

Substituting the values:

\[
171 - 79 = 92
\]

So, **92 students** studied only in the library.

### Part B: Number of students who studied only in the cafeteria
Similarly, students who studied only in the cafeteria can be calculated by subtracting those who studied in both places from those who studied in the cafeteria.

\[
\text{Students who studied only in the cafeteria} = |C| - |L \cap C|
\]

Substituting the values:

\[
131 - 79 = 52
\]

So, **52 students** studied only in the cafeteria.

### Part C: Number of students who did not study in either of these places
To find the number of students who did not study in either the library or the cafeteria, we can first calculate the number of students who studied in at least one of these places, and then subtract it from the total number of students surveyed.

The number of students who studied in at least one place is given by:

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

Substituting the values:

\[
|L \cup C| = 171 + 131 - 79 = 223
\]

So, the number of students who did not study in either place:

\[
\text{Students who did not study in either place} = n - |L \cup C|
\]

\[
250 - 223 = 27
\]

So, **27 students** did not study in either the library or the cafeteria.

### Summary:
- **A) 92 students** studied only in the library.
- **B) 52 students** studied only in the cafeteria.
- **C) 27 students** did not study in either the library or the cafeteria.

Would you like more details on any part of this solution, or have any other questions?

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Here are 5 related questions for further practice:

1. If 50 more students were surveyed, and 30 of them studied in both places, how would the results change?
2. What is the probability that a randomly chosen student studied in either the library or the cafeteria?
3. How many students studied in only one of the two places?
4. If the number of students who studied in both places increased by 10, what would be the new total of students who did not study in either place?
5. What percentage of students studied in the library but not in the cafeteria?

**Tip:** When dealing with sets and overlaps, the principle of inclusion-exclusion is a valuable tool to accurately count elements in different groups without double-counting.

Explore a detailed analysis of survey results from a community college, focusing on where students study on campus. Learn how to calculate students who studied only in the library, only in the cafeteria, and those who did not study in either place. Suitable for high school students studying set theory and principles of inclusion-exclusion.

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