To solve this problem, let’s determine the sample space for choosing two students out of three (Patty, Quinlan, and Rashad) where the first selected student becomes the president and the second becomes the vice president.
The choices given in the problem are:
1. [tex]\( S = \{ PQR \} \)[/tex]
2. [tex]\( S = \{P Q R, P R Q, Q P R, Q R P, R P Q, R Q P\} \)[/tex]
3. [tex]\( S = \{P Q, P R, Q R\} \)[/tex]
4. [tex]\( S = \{P Q, Q P, P R, R P, Q R, R Q\} \)[/tex]
Let’s analyze each choice:
1. [tex]\( S = \{ PQR \} \)[/tex]:
- This choice represents a single combination of the three names in one specific order, which does not match the event of choosing two students with designated roles.
2. [tex]\( S = \{P Q R, P R Q, Q P R, Q R P, R P Q, R Q P\} \)[/tex]:
- This choice lists all possible permutations of the three students. This is incorrect because our sample space should only contain pairs, not full permutations involving all three students.
3. [tex]\( S = \{P Q, P R, Q R\} \)[/tex]:
- This choice lists combinations without regard to order, implying that [tex]\((P, Q)\)[/tex] is the same as [tex]\((Q, P)\)[/tex]. However, for this event, the order matters because the first selected student is the president and the second is the vice president.
4. [tex]\( S = \{P Q, Q P, P R, R P, Q R, R Q\} \)[/tex]:
- This choice correctly shows all ordered pairs of students. Each pair represents the scenarios where the first student is the president and the second is the vice president.
Hence, the correct sample space, [tex]\( S \)[/tex], for this event is:
[tex]\[S = \{P Q, Q P, P R, R P, Q R, R Q\}\][/tex]
Thus, the answer is the fourth option:
[tex]\[S=\{P Q, Q P, P R, R P, Q R, R Q\}\][/tex]
