Question

four runners, fran, olona, 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? \\( \bigcirc \\ s = \\{foi, ofi, ifo\\} \\) \\( \bigcirc \\ s = \\{foi, fio, ofi, oif\\} \\) \\( \bigcirc \\ s = \\{foi, fio, ofi, oif, ifo, iof\\} \\) \\( \bigcirc \\ s = \\{foi, fio, ofi, gif, hfo, hgi, ifo, iof\\} \\)

Explanation:

Step1: Identify the runners

The other three runners are Fran (F), Olona (O), and Imani (I). We need to find all permutations of these three.

Step2: Calculate permutations

The number of permutations of \( n \) distinct objects is \( n! \). Here, \( n = 3 \), so \( 3! = 3\times2\times1=6 \).

Step3: List all permutations

The permutations of F, O, I are:

• F first: FOI, FIO
• O first: OFI, OIF
• I first: IFO, IOF

Combining these, we get the set \( \{FOI, FIO, OFI, OIF, IFO, IOF\} \).

Answer:

\( S = \{FOI, FIO, OFI, OIF, IFO, IOF\} \) (the third option)