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Patty, Quinlan, and Rashad want to be club officers. The teacher 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, S, for this event?

A. [tex]\( S = \{P, Q, R\} \)[/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 sample space for this event, we need to consider the fact that the order in which the names are drawn matters, as the first student chosen becomes the president and the second student becomes the vice president.

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

We need to find all possible outcomes when choosing two students out of the three in a specific order. Let's list these outcomes:

1. Patty first, Quinlan second: PQ
2. Patty first, Rashad second: PR
3. Quinlan first, Patty second: QP
4. Quinlan first, Rashad second: QR
5. Rashad first, Patty second: RP
6. Rashad first, Quinlan second: RQ

Thus, the sample space, [tex]\( S \)[/tex], which includes all possible pairs in the specified order, is:

[tex]\[ S = \{PQ, QP, PR, RP, QR, RQ\} \][/tex]

So, the correct choice representing the sample space is:

[tex]\[ S = \{PQ, QP, PR, RP, QR, RQ\} \][/tex]

Hence, the correct answer is:

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

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