At a community​ college, a survey was taken to determine where students study on campus. Of the 240 students​ surveyed, it was determined that 
166 studied in the library.
132 studied in the cafeteria.
79 studied in both the library and the cafeteria.
Complete parts ​a) through ​c).
Question content area bottom
Part 1
​a) Of those students​ surveyed, how many studied only in the​ library?
  
enter your response here ​(Type a whole​ number.)
Part 2
​b) Of those students​ surveyed, how many studied only in the​ cafeteria?
  
enter your response here ​(Type a whole​ number.)
Part 3
​c) Of those students​ surveyed, how many did not study in either of these​ places?
  
enter your response here ​(Type a whole​ number.)

We can solve this problem using the principle of set theory with two overlapping sets: the students who study in the library (set L) and the students who study in the cafeteria (set C). The overlap represents students who study in both places (set \( L \cap C \)).

Let’s define:

- \( L = 166 \) (students who study in the library)
- \( C = 132 \) (students who study in the cafeteria)
- \( L \cap C = 79 \) (students who study in both)

### Part 1: How many students studied only in the library?

Students who studied only in the library would be those in the library but not in the cafeteria. This is given by:

\[
L - (L \cap C) = 166 - 79 = 87
\]

Thus, **87 students studied only in the library**.

### Part 2: How many students studied only in the cafeteria?

Similarly, students who studied only in the cafeteria would be those in the cafeteria but not in the library. This is given by:

\[
C - (L \cap C) = 132 - 79 = 53
\]

Thus, **53 students studied only in the cafeteria**.

### Part 3: How many students did not study in either the library or the cafeteria?

The total number of students who studied in either the library, the cafeteria, or both can be found by the formula for the union of two sets:

\[
|L \cup C| = L + C - (L \cap C) = 166 + 132 - 79 = 219
\]

Now, the number of students who did not study in either place is the total number of surveyed students minus those who studied in either place:

\[
240 - |L \cup C| = 240 - 219 = 21
\]

Thus, **21 students did not study in either the library or the cafeteria**.

### Final Answers:
- a) 87 students studied only in the library.
- b) 53 students studied only in the cafeteria.
- c) 21 students did not study in either place.

Would you like more details or have any other questions?

### Related questions:
1. How many students studied in either the library or the cafeteria, but not both?
2. What is the probability that a randomly selected student studies in both the library and the cafeteria?
3. What percentage of students studied only in one of the two locations?
4. If the number of students increases to 300, how would this affect the calculations?
5. How would you calculate the number of students who studied in neither place if the overlap number changes to 85?

#### Tip: When solving set theory problems, drawing a Venn diagram can be helpful in visualizing the relationships between different groups.

At a community​ college, a survey was taken to determine where students study on campus. Of the 240 students​ surveyed, it was determined that 166 studied in the library, 132 studied in the cafeteria, 79 studied in both the library and the cafeteria. Complete parts ​a) through ​c). Part 1 ​a) Of those students​ surveyed, how many studied only in the​ library? Part 2 ​b) Of those students​ surveyed, how many studied only in the​ cafeteria? Part 3 ​c) Of those students​ surveyed, how many did not study in either of these​ places?

This problem involves a set theory survey of 240 community college students, with 166 studying in the library, 132 in the cafeteria, and 79 in both. It requires calculating the number of students studying exclusively in each location and those who didn't study in either. Suitable for Grades 10-12.

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