Let's solve the problem step by step to determine the sample space [tex]\( S \)[/tex] for Carla choosing two out of her three pairs of sneakers (A, B, C) to take to a track meet.
1. Understand the Requirement
- We need to find all possible combinations of choosing 2 out of 3 sneakers.
- A combination is a selection of items where the order does not matter.
2. Identify Possible Pairs
- Since the order does not matter, we do not consider permutations (i.e., AB is the same as BA).
3. List All Combinations
- Here are the possible pairs Carla can make:
- Pair 1: A and B
- Pair 2: A and C
- Pair 3: B and C
Therefore, the sample space [tex]\( S \)[/tex] should list these three pairs: AB, AC, and BC.
So, the correct choice representing the sample space for Carla's selection is:
[tex]\[ S = \{AB, AC, BC\} \][/tex]
Thus, the sample space [tex]\( S \)[/tex] for the event is:
[tex]\[ S = \{A B, A C, B C\} \][/tex]
The correct answer is:
[tex]\[ S = \{A B, A C, B C\} \][/tex]
