Answer :
To determine which choice represents the sample space \( S \) for the event, we need to identify all the possible combinations of 3 students taken from a group of 4 students (Ariana, Boris, Cecile, and Diego).
Let's list all the possible combinations of 3 students:
1. Choosing Ariana, Boris, and Cecile: \( \{A, B, C\} \)
2. Choosing Ariana, Boris, and Diego: \( \{A, B, D\} \)
3. Choosing Ariana, Cecile, and Diego: \( \{A, C, D\} \)
4. Choosing Boris, Cecile, and Diego: \( \{B, C, D\} \)
The sample space \( S \) is the set of these combinations. There are no other unique combinations of 3 students from the set of 4 students. Therefore, we need to find the choice that lists these combinations correctly without duplications or permutations.
The given choices are:
1. \( S = \{ABC, ABD, ACD, BCD\} \)
2. \( S = \{ABC, BCA, CDA, DAC\} \)
3. \( S = \{ABC, ABD, BAC, BDA, ACD, ADC, BCD, BDC\} \)
4. \( S = \{ABC, ABD, ACD, BCA, BCD, BDC, CAB, CBD, DAC, DBC\} \)
Comparing each choice:
1. \( \{ABC, ABD, ACD, BCD\} \)
- Contains exactly the unique combinations we listed: \( \{A, B, C\}, \{A, B, D\}, \{A, C, D\}, \{B, C, D\} \).
2. \( \{ABC, BCA, CDA, DAC\} \)
- Contains permutations of the same set of students which is incorrect.
3. \( \{ABC, ABD, BAC, BDA, ACD, ADC, BCD, BDC\} \)
- Contains duplicate sets with the same students reordered which is incorrect.
4. \( \{ABC, ABD, ACD, BCA, BCD, BDC, CAB, CBD, DAC, DBC\} \)
- Contains several unnecessary permutations and duplicates which are incorrect.
Thus, the correct choice that represents the sample space \( S \) for this event is:
```
1. \( S = \{ABC, ABD, ACD, BCD\} \)
```
Let's list all the possible combinations of 3 students:
1. Choosing Ariana, Boris, and Cecile: \( \{A, B, C\} \)
2. Choosing Ariana, Boris, and Diego: \( \{A, B, D\} \)
3. Choosing Ariana, Cecile, and Diego: \( \{A, C, D\} \)
4. Choosing Boris, Cecile, and Diego: \( \{B, C, D\} \)
The sample space \( S \) is the set of these combinations. There are no other unique combinations of 3 students from the set of 4 students. Therefore, we need to find the choice that lists these combinations correctly without duplications or permutations.
The given choices are:
1. \( S = \{ABC, ABD, ACD, BCD\} \)
2. \( S = \{ABC, BCA, CDA, DAC\} \)
3. \( S = \{ABC, ABD, BAC, BDA, ACD, ADC, BCD, BDC\} \)
4. \( S = \{ABC, ABD, ACD, BCA, BCD, BDC, CAB, CBD, DAC, DBC\} \)
Comparing each choice:
1. \( \{ABC, ABD, ACD, BCD\} \)
- Contains exactly the unique combinations we listed: \( \{A, B, C\}, \{A, B, D\}, \{A, C, D\}, \{B, C, D\} \).
2. \( \{ABC, BCA, CDA, DAC\} \)
- Contains permutations of the same set of students which is incorrect.
3. \( \{ABC, ABD, BAC, BDA, ACD, ADC, BCD, BDC\} \)
- Contains duplicate sets with the same students reordered which is incorrect.
4. \( \{ABC, ABD, ACD, BCA, BCD, BDC, CAB, CBD, DAC, DBC\} \)
- Contains several unnecessary permutations and duplicates which are incorrect.
Thus, the correct choice that represents the sample space \( S \) for this event is:
```
1. \( S = \{ABC, ABD, ACD, BCD\} \)
```
