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Four speed skaters, Marco, Naim, Oliver, and Pedro, compete in a relay race where they all agree that Naim will be the last skater. They try to decide whether or not Oliver should be the first skater in the race.

Which subset, [tex]A[/tex], of the sample space shows the complement of the event in which Oliver is the first skater in the race?

A. [tex]A=\{ \text{MOPN, PMON} \}[/tex]

B. [tex]A=\{ \text{OPMN, OMPN} \}[/tex]

C. [tex]A=\{ \text{MOPN, MPON, OPMN, POMN} \}[/tex]

D. [tex]A=\{ \text{MOPN, MPON, PMON, POMN} \}[/tex]

Sagot :

GinnyAnsweridn
Given four speed skaters, Marco, Oliver, Pedro, and Naim, it's agreed that Naim will be the last skater in the race. We need to find the complement of the event where Oliver is the first skater in the race.

First, we list all possible orders where Naim is the last skater:
- MOPN
- MONP
- MPO N
- MPNO
- OMPN
- OMNP
- OPMN
- OPNM
- ONMP
- ONPM
- PMON
- PMNO
- PNOM
- PONM
- POMN
- PONM

Next, we identify the permutations where Oliver is the first skater:
- OMPN
- OMNP
- OPMN
- OPNM
- ONMP
- ONPM

The complement of this event is the set of all permutations without Oliver being the first skater. Therefore, we remove all permutations where Oliver is the first skater from the list. This leaves us with:

- MOPN
- MONP
- MPON
- MPNO
- PMON
- PMNO
- PNOM
- PONM
- POMN
- PONM

So, the subset [tex]\( A \)[/tex] of the sample space that shows the complement of the event in which Oliver is the first skater in the race is:

[tex]\[ A = \{ MOPN, MONP, MPON, MPNO, PMON, PMNO, PNOM, PONM, POMN \} \][/tex]

Comparing this to the provided options:
[tex]\[ A = \{MOPN, MPON, PMON, POMN\} \][/tex] matches correctly.

Therefore, the correct answer is:
[tex]\[ A=\{MOPN, MPON, PMON, POMN\} \][/tex]
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