Let's solve this problem step-by-step:
1. Identify the participants: The students are Patty (P), Quinlan (Q), and Rashad (R).
2. Determine the event: The event involves choosing two students from the three participants. The first chosen student will be the president, and the second chosen student will be the vice president.
3. Establish the order significance: Since the first student chosen is the president and the second is the vice president, the order in which the students are chosen matters.
4. List all possible outcomes: Let's list all the possible ways we can select two students from the three participants, considering the order:
- If Patty is chosen first, Quinlan or Rashad can be chosen second: (P, Q), (P, R).
- If Quinlan is chosen first, Patty or Rashad can be chosen second: (Q, P), (Q, R).
- If Rashad is chosen first, Patty or Quinlan can be chosen second: (R, P), (R, Q).
5. Write the sample space: The sample space should include all unique outcomes listed above:
[tex]\[
S = \{(P, Q), (Q, P), (P, R), (R, P), (Q, R), (R, Q)\}
\][/tex]
6. Match the choices given: Comparing the sample space we derived with the given choices:
[tex]\[
\text{Choice D: } S = \{(P, Q), (Q, P), (P, R), (R, P), (Q, R), (R, Q)\}
\][/tex]
Thus, the correct choice representing the sample space [tex]\( S \)[/tex] for this event is:
[tex]\[
S = \{(P, Q), (Q, P), (P, R), (R, P), (Q, R), (R, Q)\}
\][/tex]
This matches the fourth choice provided:
[tex]\[
S = \{P Q, Q P, P R, R P, Q R, R Q\}
\][/tex]
Therefore, the correct answer is:
[tex]\[ S = \{P Q, Q P, P R, R P, Q R, R Q\} \][/tex]
