To solve this problem, we need to determine all possible orders of the other three runners: Fran (F), Gkira (G), and Imani (I), while keeping in mind that Hakey (H) is always the first runner and does not participate in these permutations.
To construct the sample space, we will generate all permutations of the set {F, G, I}. In permutations, the order matters, which means that each unique arrangement of F, G, and I represents a different permutation.
Let's list out all possible permutations of the three runners Fran (F), Gkira (G), and Imani (I):
1. F, G, I
2. F, I, G
3. G, F, I
4. G, I, F
5. I, F, G
6. I, G, F
Therefore, the complete sample space \( S \) showing all the possible orders of Fran, Gkira, and Imani is:
[tex]\[ S = \{FGI, FIG, GFI, GIF, IFG, IGF\} \][/tex]
So the correct answer is:
[tex]\[ S = \{F G I, F I G, G F I, G I F, I F G, I G F\} \][/tex]
