If a runner finishes 5th, what possible ways can 1st, 2nd, and 3rd place be filled?

To determine the number of possible ways to fill 1st, 2nd, and 3rd places when a runner finishes 5th, we need to consider how many other runners are competing. The finish position 5th indicates that there are at least four other runners who could occupy the top three positions.

Assuming there are a total of five runners, the number of ways to choose and arrange the runners in the 1st, 2nd, and 3rd positions can be calculated using the concept of permutations. Specifically, this can be calculated by determining how many ways we can select three runners from the available pool (the four other runners since the runner who finished fifth cannot be in the top three).

The formula for the number of permutations of selecting and arranging 'r' items from a total of 'n' items is given by:

[ P(n, r) = \frac{n!}{(n - r)!} ]

For this problem:

• 'n' is 4 (the four runners who can be 1st, 2nd, or 3rd)

• 'r' is 3 (the positions we need to fill)

So the calculation becomes:

[ P(4,