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There are 15 gymnasts in the Olympic women's competition, including 4 Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded to three of the 15 gymnasts, if exactly one of the Americans wins a medal?

 Mar 20, 2020
 #1
avatar+111360 
+2

Either the American wins the gold, silver or bronze   (3 ways)

 

And in each case, we want to choose 1 of 4 Americans to win one of the medals  and  any 2 of the other 11 to receive the other two medals

 

So...we have

 

3 * C(4,1) * ( 11, 2)   = 

 

660  ways

 

 

cool cool cool

 Mar 20, 2020
 #2
avatar+2 
+1

CPHill, you got it wrong. Try using casework.

 Mar 25, 2020
 #3
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+1

The problem is that the 4 americans each have a possibility of winning a different metal, so the total number of combinations that an american could win a metal.

 

4*3

 

or 4 americans that can win 3 different metals. Now the other two don't matter as much so we take the 11 other contestants for one of them, and then we have 10 left to chose from. So the answer is 

 

4 * 3 * 11 * 10 = 1320

 Mar 26, 2020

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