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?

Answer 1

There are three different choices for the American.

There are 16 different choices for the first non-American and 15 different choices for the other non-American.

So, there are 3 x 16 x 15 different groups.

But, order is important -- who gets the gold, who gets the silver, and who gets the bronze -- there are 6 different ways to arrange the 3 winners (3! = 6).

Thus, there are 6 x 3 x 16 x 15 different results.