There are 8 sprinters in the Olympic 100-meter finals. Three of the sprinters are Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if at most one American gets a medal?
Here's my best attempt.....
Consider the sets where one of the Americans wins gold...we have 3 possibilities for the American gold medal
And of the other 5 sprinters, we want to choose any of the other 5 non-Americans to receive the silver and any 4 of the non-Americans to recieve the bronze
So...this gives 3 * 5 * 4 = 60 possible ways
And we will have the same number of sets for an American winning silver and the same number of sets for an American winning bronze
So far... this gives 60 * 3 = 180 ways
To this..we must add the sets where no American wins any medals....we have 5 ways to award the gold medal, 4 ways to award the silver and 3 ways to award the bronze = 5 * 4 * 3 = 60 ways
So 180 + 60 = 240 ways where, at most, one American wins a medal