Three different medals- Gold, Silver and Bronze- are awarded to athletes in two different races. If no athlete may win more than one medal, and there are 6 athletes in total, how many different combinations are there?

I know that the first race: $6\cdot 5\cdot 4= 120$

I feel like the answer for both races should be $120+120= 240$.

But this is not an answer choice so I must be wrong!


1 Answer 1


I think the text is trying to say that no athlete can win two medals in both races, thus every participant (since there’s $6$ of them and $6$ medals) must win one and only one medal.

So the answer is just the number of ways you gan give these $6$ medals to the $6$ racers, or simply $6!$

  • $\begingroup$ Thank you for your help! That is what I first thought, but the answer choices are: 36, 72, 180, 300 or 720 so I think the text means: all 6 racers are running in both races and each of the 6 can win a medal in either race, but nobody can win two medals in the same race (duh)! $\endgroup$
    – Looney
    Jun 9, 2019 at 19:18
  • $\begingroup$ Oh, wait. 720 is (6!) so maybe that IS what it means!!!!! What a poorly written math question. $\endgroup$
    – Looney
    Jun 9, 2019 at 19:21
  • $\begingroup$ Glad to help! And yeah, I suppose it could’ve been phrased a little better... $\endgroup$
    – user622002
    Jun 9, 2019 at 20:22

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