Four swimmers, Daniela, Camille, Brennan, and Amy, compete on a relay team. For the first race of the year, Daniela begins the relay. The other three swimmers can swim in any order. The sample space, S, for the event is shown below.

Question

S = {CBA, CAB, BAC, BCA, ACB, ABC}

After the first race, it is determined that Camille is a strong finisher and should be the final swimmer in the race.

What subset, A, of the sample space represents the complement of the event in which Camille is the final swimmer?

A = {CBA, CAB, BCA, ACB}
A = {ABC, BAC}
A = {CBA, CAB, BAC, BCA, ACB, ABC}
A = {AB, BA}

Answer 1

Given that the sample space \( S \) is:

\[ S = {CBA, CAB, BAC, BCA, ACB, ABC} \]

we are looking for the subset \( A \) that represents the complement of the event in which Camille is the final swimmer.

To find the swimmers' combinations where Camille is the final swimmer, we should check each element of \( S \).

The combinations with Camille (C) in the final position are:

• CBA
• CAB

So, the remaining combinations where Camille is NOT the final swimmer are:

• BAC
• BCA
• ACB
• ABC

Thus, the complement of the event where Camille is the final swimmer is:

\[ A = {BAC, BCA, ACB, ABC} \]

This does not match any of the options provided; therefore, none of those subsets represent the correct complement properly.

Given that the question is asking for the complement of the event where Camille is the final swimmer, none of the options \( A = {CBA, CAB, BCA, ACB} \), \( A = {ABC, BAC} \), \( A = {CBA, CAB, BAC, BCA, ACB, ABC} \), or \( A = {AB, BA} \) align with our finding.

In summary, the answer should include all instances where Camille is NOT the final swimmer:

\[ A = {BAC, BCA, ACB, ABC} \]

Since none of the provided options accurately represent this conclusion, the query highlights an inconsistency.