Four runners, Fran, Gloria, Haley, and Imani, compete on a relay team. Haley is the first runner in the relay. The other runners can run in any order.
What is the sample space showing the possible orders of the other three runners?
A. [tex]$S =\{ FGI , GFI , IFG \}$[/tex]
B. [tex]$S =\{ FGI , FIG , GFI , GIF \}$[/tex]
C. [tex]$S =\{ FGI , FIG , GFI , GIF , IFG , IGF \}$[/tex]
D. [tex]$S =\{ FGI , FIG , GFI , GIF , HFG , HGI , IFG , IGF \}$[/tex]
Answer (2)
List all possible orders of the three runners: Fran, Gloria, and Imani.
The possible orders are: FGI, FIG, GFI, GIF, IFG, IGF.
The sample space S includes all these permutations.
The sample space is: { FG I , F I G , GF I , G I F , I FG , I GF }
Explanation
Analyze the problem We need to determine all possible orders in which the remaining three runners (Fran, Gloria, and Imani) can run after Haley. This is a permutation problem, as the order matters.
List all possible orders Let's list all possible orders:
Fran, Gloria, Imani (FGI)
Fran, Imani, Gloria (FIG)
Gloria, Fran, Imani (GFI)
Gloria, Imani, Fran (GIF)
Imani, Fran, Gloria (IFG)
Imani, Gloria, Fran (IGF)
Define the sample space Therefore, the sample space S is the set of all these permutations:
S = { FG I , F I G , GF I , G I F , I FG , I GF }
Final Answer The sample space showing the possible orders of the other three runners is { FG I , F I G , GF I , G I F , I FG , I GF } .
Examples
In a race, understanding the possible finishing orders helps coaches plan strategies. For example, if you have 3 runners, knowing all 6 possible finish orders (permutations) allows you to analyze the likelihood of different team outcomes and optimize runner placement to maximize the team's overall performance. The number of possible arrangements can be calculated using the factorial: 3 ! = 3 × 2 × 1 = 6 .
The sample space of possible orders for the runners Fran, Gloria, and Imani is { FGI , FIG , GFI , GIF , IFG , IGF }. This indicates all the unique arrangements for the three runners after Haley. Thus, the correct answer is option C.
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