To solve this problem, let's define the possible choices Carla has when selecting two pairs of sneakers out of three available pairs named A, B, and C.
### Step-by-step Solution:
1. Identify the pairs of sneakers: Carla has three pairs of sneakers named [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
2. Determine the possible combinations: Carla needs to choose 2 pairs out of the 3 available. When choosing combinations, the order does not matter, so [tex]\( (A, B) \)[/tex] is the same as [tex]\( (B, A) \)[/tex].
3. List all combinations:
- Pair [tex]\( A \)[/tex] with pair [tex]\( B \)[/tex] to get [tex]\( (A, B) \)[/tex].
- Pair [tex]\( A \)[/tex] with pair [tex]\( C \)[/tex] to get [tex]\( (A, C) \)[/tex].
- Pair [tex]\( B \)[/tex] with pair [tex]\( C \)[/tex] to get [tex]\( (B, C) \)[/tex].
4. Form the sample space, [tex]\( S \)[/tex]: The sample space is the set of all possible combinations of 2 pairs Carla can choose from her 3 pairs of sneakers.
Therefore, the sample space [tex]\( S \)[/tex] is:
[tex]\[
S = \{(A, B), (A, C), (B, C)\}
\][/tex]
5. Review the answer choices:
- [tex]\( S = \{A B C\} \)[/tex]: This option incorrectly lists the entire set of sneakers together, not the combinations.
- [tex]\( S = \{A B C, C A B\} \)[/tex]: This option incorrectly lists permutations of all three sneakers.
- [tex]\( S = \{A B, A C, B C\} \)[/tex]: This option correctly lists the combinations of choosing 2 pairs out of 3 without repetition or regard for order.
- [tex]\( S = \{A B, B A, A C, C A, B C, C B\} \)[/tex]: This option lists permutations of pairs, which does not apply since order does not matter in combinations.
### Conclusion:
The correct choice that represents the sample space [tex]\( S \)[/tex] for the event where Carla chooses 2 out of 3 pairs of sneakers is:
[tex]\[
S = \{A B, A C, B C\}
\][/tex]
So, the third option is the correct answer.
