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

To determine the number of ways that 1st, 2nd, and 3rd places can be filled given that a runner has finished in 5th place, we can analyze the situation in terms of permutations.

Since we have a total of 5 runners, and one of them (the runner in 5th place) is already accounted for, there are 4 remaining runners who can potentially finish in the top three spots. The key is that we can select any 3 runners from these 4 and arrange them in the top three positions.

For the 1st place, we have 4 choices since any of the 4 runners can finish first. Once a runner is chosen for 1st place, there are 3 runners left who can finish in 2nd place. After that selection, there are 2 remaining runners available to fill the 3rd place.

The total number of combinations to select and arrange the runners in the top three spots is calculated as follows:

1. Choose a runner for 1st place: 4 options

2. Choose a runner from the remaining three for 2nd place: 3 options

3. Choose a runner from the remaining two for 3rd place