Question

four speed skaters, marco, naim, oliver, and pedro, compete in a relay race where they all agree that naim will be the last skater. then, they try to decide whether or not oliver should be the first skater in the race. which subset, a, of the sample space shows the complement of the event in which oliver is the first skater in the race? a = {mopn, pmon} a = {opmn, ompn} a = {mopn, mpon, opmn, pomn} a = {mopn, mpon, pmon, pomn}

Explanation:

Step1: Define fixed position

Naim (N) is fixed as 4th. We arrange Marco (M), Oliver (O), Pedro (P) in first 3 spots.

Step2: Find target event set

Event: Oliver is first. Fix O in 1st spot, arrange M/P in 2nd/3rd: {OPMN, OMPN}

Step3: Find total sample space

All valid arrangements: {OPMN, OMPN, MOPN, MPON, PMON, POMN} (6 total)

Step4: Calculate complement set

Subtract target event from total space:
$\text{Complement} = \text{Total} - \text{Oliver first} = \{\text{MOPN, MPON, PMON, POMN}\}$

Answer:

A = {MOPN, MPON, PMON, POMN}