Let's consider the problem step-by-step to find the correct answer. We know that Haley must run first in the relay, and that leaves us with Fran, Gloria, and Imani to organize in the remaining positions.
To find the sample space showing the possible orders of these three runners (Fran, Gloria, and Imani), we need to determine all possible permutations of the three runners.
The possible permutations of the three runners (Fran, Gloria, and Imani) are:
1. F, G, I
2. F, I, G
3. G, F, I
4. G, I, F
5. I, F, G
6. I, G, F
Since Haley always has to be first, we will prepend 'H' to each of these permutations:
1. HFGI
2. HFIG
3. HGFI
4. HGIF
5. HIFG
6. HIGF
This results in the following possible orders for the runners, taking into account Haley running first and the other three runners following in all possible orders:
[tex]\[ S = \{HFGI, HFIG, HGFI, HGIF, HIFG, HIGF\} \][/tex]
Thus, the correct sample space showing the possible orders of all four runners, with Haley fixed as the first runner, is:
[tex]\[ S = \{F G I, F I G, G F I, G I F, I F G, I G F\} \][/tex]
This corresponds with the answer choice:
[tex]\[ S = \{F G I, F I G, G F I, G I F, ~ I F G, ~ I G F\} \][/tex]
