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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]



Answer :

GinnyAnswer
To find the complement of the event in which Oliver is the first skater in the race, we need to determine all possible orders of the skaters complying with the condition that Naim is the last skater but Oliver is not the first skater.

Given that the skaters must end with Naim, the possible orders can be analyzed as follows:

1. Identify all possible orders without Oliver being the first skater:
- Marco, Oliver, Pedro, Naim
- Marco, Pedro, Oliver, Naim
- Pedro, Marco, Oliver, Naim
- Pedro, Oliver, Marco, Naim

Therefore, the subsets where Oliver is not the first skater and Naim is the last skater are:
1. MOPN
2. MPON
3. PMON
4. POMN

Thus, the subset [tex]\(A\)[/tex] that represents the complement of Oliver being the first skater is:
[tex]\[A = \{\text{MOPN, MPON, PMON, POMN}\}\][/tex]

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

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