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The teacher who directs the club will place the names of the club officers 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]\( S = \{P, Q, R\} \)[/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 choosing a president and a vice president from a group of students, we need to consider the order of selection since the first name chosen will be president and the second will be vice president. Let's list the possibilities:

1. First choice, [tex]$P$[/tex] and second choice, [tex]$Q$[/tex]:
- President: [tex]$P$[/tex]
- Vice President: [tex]$Q$[/tex]
- Ordered pair: [tex]$(P, Q)$[/tex]

2. First choice, [tex]$P$[/tex] and second choice, [tex]$R$[/tex]:
- President: [tex]$P$[/tex]
- Vice President: [tex]$R$[/tex]
- Ordered pair: [tex]$(P, R)$[/tex]

3. First choice, [tex]$Q$[/tex] and second choice, [tex]$P$[/tex]:
- President: [tex]$Q$[/tex]
- Vice President: [tex]$P$[/tex]
- Ordered pair: [tex]$(Q, P)$[/tex]

4. First choice, [tex]$Q$[/tex] and second choice, [tex]$R$[/tex]:
- President: [tex]$Q$[/tex]
- Vice President: [tex]$R$[/tex]
- Ordered pair: [tex]$(Q, R)$[/tex]

5. First choice, [tex]$R$[/tex] and second choice, [tex]$P$[/tex]:
- President: [tex]$R$[/tex]
- Vice President: [tex]$P$[/tex]
- Ordered pair: [tex]$(R, P)$[/tex]

6. First choice, [tex]$R$[/tex] and second choice, [tex]$Q$[/tex]:
- President: [tex]$R$[/tex]
- Vice President: [tex]$Q$[/tex]
- Ordered pair: [tex]$(R, Q)$[/tex]

These combinations account for all possible ways of selecting a president and a vice president from three students, considering the order of selection is crucial.

Therefore, the sample space [tex]\( S \)[/tex] that represents all possible outcomes is:
[tex]\[ S = \{(P, Q), (Q, P), (P, R), (R, P), (Q, R), (R, Q)\} \][/tex]

This matches the fourth choice given, where the order of selection is considered:
[tex]\[ S = \{P Q, Q P, P R, R P, Q R, R Q\} \][/tex]

Thus, the correct choice that represents the sample space [tex]\( S \)[/tex] is:
[tex]\[ S=\{P Q, Q P, P R, R P, Q R, R Q\} \][/tex]

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