Let's carefully determine the sample space for drawing two names from the hat and assigning roles of president and vice president.
We have three students: Patty (P), Quinlan (Q), and Rashad (R). The teacher will draw two names from the hat in sequence. The first name drawn becomes the president, and the second name drawn becomes the vice president.
We need to consider all possible ordered pairs of the students that can be drawn. Each ordered pair represents one possible outcome of this event, where the order matters (since the first name drawn gets a different role than the second name drawn).
Let's analyze the possible outcomes:
1. If Patty (P) is drawn first, we can have:
- (P, Q): Patty is president, Quinlan is vice president.
- (P, R): Patty is president, Rashad is vice president.
2. If Quinlan (Q) is drawn first, we can have:
- (Q, P): Quinlan is president, Patty is vice president.
- (Q, R): Quinlan is president, Rashad is vice president.
3. If Rashad (R) is drawn first, we can have:
- (R, P): Rashad is president, Patty is vice president.
- (R, Q): Rashad is president, Quinlan is vice president.
So, the sample space for this event, considering these ordered pairs, is:
[tex]\[ S = \{ (P, Q), (P, R), (Q, P), (Q, R), (R, P), (R, Q) \} \][/tex]
This represents all the possible ways two names can be drawn from the hat in sequence and assigned the roles of president and vice president.
Therefore, the correct representation of the sample space is:
[tex]\[ S = \{ (P, Q), (Q, P), (P, R), (R, P), (Q, R), (R, Q) \} \][/tex]
So, the correct answer is:
[tex]\[ S = \{P Q, Q P, P R, R P, Q R, R Q\} \][/tex]
