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In a knockout tournament 2^(n) equally s...

In a knockout tournament `2^(n)` equally skilled players, `S_(1),S_(2),….S_(2n),` are participatingl. In each round, players are divided in pair at random and winner from each pair moves in the next round. If `S_(2)` reaches the semi-final, then the probability that `S_(1)` wins the tournament is 1/84. The value of n equals _______.

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To solve the problem, we need to find the value of \( n \) in a knockout tournament with \( 2^n \) equally skilled players, where the probability that player \( S_1 \) wins the tournament given that player \( S_2 \) reaches the semi-finals is \( \frac{1}{84} \). ### Step-by-step Solution: 1. **Understanding the Tournament Structure**: - In a knockout tournament with \( 2^n \) players, there will be \( n \) rounds (since each round halves the number of players). - The semi-finals will consist of 4 players, which means \( n \) must be at least 2 (since \( 2^2 = 4 \)). ...
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