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I would like to confirm if my answer is correct for the following question:

A conventional knock-out tournament begins with $2^n$ competitors and has $n$ rounds. There are no play-offs for the positions $2,3,..,2^{n-1}$, and the initial table of draws is specified. Give a concise description of the sample space of all possible outcomes.

My answer is: $$\sum_{i=1}^{n} 2^{2^{i}/2}.$$

My reasoning is as follows: For each round, the number of players is $2^n$. The number of matches for each round is $2^n / 2$. Since there can be only a win or loss, the total number of combinations per round is $2^{2^{n}/2}$. Therefore the total number of combinations of outcomes (the total sample space), is the sum of combinations in each round from $1 \ldots n$.

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What is the variable in the summation? –  vadim123 Feb 18 at 17:58
Corrected the summation. –  I.K. Feb 18 at 18:00
"For each round, the number of players is $2^n$": No it's not. –  TonyK Feb 18 at 18:04

1 Answer 1

up vote 1 down vote accepted

You don't need to sum any complicated series here. Simply observe that each match eliminates one competitor, therefore there are $2^n-1$ matches, each of which has two possible outcomes.

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No! Much more than that! –  TonyK Feb 18 at 18:38
Answer is: $2^{2^{n} - 1}$. –  I.K. Feb 18 at 18:52

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