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]
