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There are 32 players in a tennis tournament which is played by elimination. The tournament bracket is determined by random draw before the first match (i.e., there is no seeding). How many possible results of the tournament are there? We do not care who played whom, just who finished in the first round, who finished in the second round, and so on.

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  • $\begingroup$ Do you mean "...., just jwho was eliminated in the first, second, etc., round"? If so, that would be equivalent to you only being interested in the final standings. $\endgroup$ – Bobson Dugnutt Oct 17 '16 at 20:19
  • $\begingroup$ @Lovsovs Basically, I'm only interested in the possibilities of the victor $\endgroup$ – Gerard L. Oct 17 '16 at 20:24
  • $\begingroup$ That still isn't clear. The victor of the whole tournament? Then there's, of course, 32 possible outcomes. How do you define a victor and a round? $\endgroup$ – Bobson Dugnutt Oct 17 '16 at 20:26
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From the $32$, pick one winner. and a silver medalist. Then pick two who were eliminated in the semi's. Then pick four who were eliminated in the quarters. Then... How many ways can this be done?

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  • $\begingroup$ So you have 2 ways to eliminate ppl to determine silver, then 3*4=12 ways to eliminate for semis, then 8*7*6*5=1680 ways to eliminate for quarters, then 16*15*14*12*11*10*9=518918400 ways to eliminate for eighth-finals, and 32*31*..*18*17 ways for 16th finals? Am I correct? $\endgroup$ – Gerard L. Oct 17 '16 at 20:34
  • $\begingroup$ @GerardL. No, you have $32$ people to choose from for gold medalist, and $31$ for silver. Then you have $30\cdot 29$ pairs of people to choose from to get eliminated in the semis. But you don't care which semi they were defeated in, so really it's $\frac{30\cdot 29}2$. And so on. $\endgroup$ – Arthur Oct 17 '16 at 20:46
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    $\begingroup$ Gold:32 Silver:31 semi: 30*29/2 quarter: 28*27*26*25/4! eighth: 24*23*22*21*20*19*18*17/8! Is that correct now? $\endgroup$ – Gerard L. Oct 17 '16 at 20:56
  • $\begingroup$ @GerardL. Exactly! $\endgroup$ – Arthur Oct 17 '16 at 20:57
  • $\begingroup$ And then you add, correct? Or do you multiply? (I think its multiply? $\endgroup$ – Gerard L. Oct 17 '16 at 20:59
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Since every game eliminates one player, there are $31$ total games in the tournament. Every game has two possibilities: either Team A wins or Team B wins. So there are $2^{31}$ possible outcomes.

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  • $\begingroup$ I think you may have misunderstood the question. $\endgroup$ – Gerard L. Oct 17 '16 at 21:11

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