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Patty, Quinlan, and Rashad want to be club officers. The teacher who directs the club will place their names in a hat and choose two without looking. The student whose name is chosen first will be president, and the student whose name is chosen second will be vice president.

Which choice represents the sample space, [tex]S[/tex], for this event?

A. [tex]S=\{P O R\}[/tex]
B. [tex]S=\{P Q R, P R Q, Q P R, Q R P, R P Q, R Q P\}[/tex]
C. [tex]S=\{P Q, P R, Q R\}[/tex]
D. [tex]S=\{P Q, Q P, P R, R P, Q R, R Q\}[/tex]



Answer :

GinnyAnswer
To determine the correct sample space for the event of choosing two names out of three (Patty, Quinlan, and Rashad) for the positions of president and vice president, we need to consider the different possible outcomes when the order of selection matters because the first student chosen will be the president and the second will be the vice president.

We have three students: Patty (P), Quinlan (Q), and Rashad (R).

Let's analyze the sample space where the sequence matters:

1. Patty is chosen first, and Quinlan is chosen second (P Q).
2. Patty is chosen first, and Rashad is chosen second (P R).
3. Quinlan is chosen first, and Patty is chosen second (Q P).
4. Quinlan is chosen first, and Rashad is chosen second (Q R).
5. Rashad is chosen first, and Patty is chosen second (R P).
6. Rashad is chosen first, and Quinlan is chosen second (R Q).

So, considering all possible ordered outcomes, the sample space [tex]\( S \)[/tex] should include all these pairs because the order in which the names are chosen makes a difference for the roles. Therefore, the correct sample space for the event is:

[tex]\[ S = \{P Q, Q P, P R, R P, Q R, R Q\} \][/tex]

Thus, the correct choice is:
[tex]\[ S=\{P Q, Q P, P R, R P, Q R, R Q\} \][/tex]

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