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# Man, Casework Counting Really hates Me...

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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?

Apr 26, 2018

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8P3=8!/5!=8*7*6=336.

Apr 26, 2018
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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

Apr 26, 2018
edited by CPhill  Apr 26, 2018
edited by CPhill  Apr 26, 2018
edited by CPhill  Apr 26, 2018
edited by CPhill  Apr 26, 2018