Consider the arrangement of the athletes: 1 2 3 4 5 6
The athletes can be considered as numbers to be arranged in 6 available slots: _ _ _ _ _ _
The first slot can be filled in 6 ways, the second slot in 5 ways and so on.
So, the total number of ways in which we can have the athletes in the slots is 6*5*4*3*2*1 = 6! = 720
Now, due to the symmetry of the arrangement, there can be exactly half the number of ways in which any one of the athletes can be above any other athlete.
This can be explained using probability. Consider the 6 slots: _ _ _ _ _ _
If we fill the slots randomly with the numbers 1 to 6, what is the probability of finding number 2 in the first slot? It would be 1/6. Similarly, what is the probability of finding the number 2 in the second slot? It is 1/6 again. And similarly, the probability of finding number 2 in the third slot is 1/6.
Hence the probability of finding the number 2 in the first OR second OR third slot is 1/6 + 1/6 + 1/6 = 3/6 = 1/2
In any of these conditions, the number 5 can occupy the fourth OR fifth OR sixth slot and hence the number 2 can be ahead or to the left of number 5 in half of the conditions or arrangement.
We apply this logic to the given question in order to get the number of possible results for the 100 meter race in which #5 finishes ahead of #2 given that there's no ties.
So in exactly half of the possible arrangements i.e. 6!/2 i.e. 720/2 = 360 results will be such that #5 finishes ahead of #2 in the race.
This logic can be applied in questions with symmetrical arrangements such as seating people, creating ranklists, etc.
Hope this helps.
MBA Candidate 2015 | Georgetown University
McDonough School of Business