There are 19 figure skaters in the Olympic women's competition, including 3 Americans. The gold medal goes to first place, silver to second, and bronze to third.

Suppose that the Olympics powers-that-be decide that exactly one American must win a medal (no more and no less). Now how many ways are there to award the medals?

Question

There are 19 figure skaters in the Olympic women's competition, including 3 Americans. The gold medal goes to first place, silver to second, and bronze to third.

Suppose that the Olympics powers-that-be decide that exactly one American must win a medal (no more and no less). Now how many ways are there to award the medals?

kropot72 · Answer

With American #1 in first place there are 16P2 permutations with non-Americans as placegetters.
With American #1 in second place there are 16P2 permutations with non-Americans as placegetters.
With American #1 in third place there are 16P2 permutations with non-Americans as placegetters.
Therefore there are 3 * 16P2 ways to award a medal to only American #1. There is the same number of ways to award a medal to only American #2 and to only American #3.

Therefore the total number of ways to award the medals with one and only one American receiving a medal is given by$$9\times 16P2=3\times \frac{16!}{14!}=you\ can\ calculate$$