In a tournament, there are \( n \) teams \( T_1, T_2, \ldots, T_n \), with \( n > 5 \). Each team consists of \( k \) players, \( k > 3 \). The following pairs of teams have one player in common: \( T_1 \) & \( T_2 \), \( T_2 \) & \( T_3 \), \( \ldots \), \( T_{n-1} \) & \( T_n \), \( T_n \) & \( T_1 \). No other pair of teams has any player in common. How many players are participating in the tournament, considering all the \( n \) teams together?
In a tournament, there are \( n \) teams \( T_1, T_2, \ldots, T_n \), with \( n > 5 \). Each team consists of \( k \) players, \( k > 3 \). The following pairs of teams have one player in common: \( T_1 \) & \( T_2 \), \( T_2 \) & \( T_3 \), \( \ldots \), \( T_{n-1} \) & \( T_n \), \( T_n \) & \( T_1 \). No other pair of teams has any player in common. How many players are participating in the tournament, considering all the \( n \) teams together?
Show Hint
- \( n(k-1) \)
- \( k(n-1) \)
- \( n(k-2) \)
- \( k(n-2) \)
- \( (n-1)(k-1) \)
The Correct Option is A
Solution and Explanation
To solve the problem, we need to determine the number of unique players participating in the tournament. We have \( n \) teams, each with \( k \) players, and pairs of teams share exactly one player according to the given pattern.
Step-by-step Explanation:
- Each team \( T_i \) has \( k \) players.
- Each pair of consecutive teams shares exactly one player. For example, teams \( T_1 \) and \( T_2 \) share one player, while teams \( T_2 \) and \( T_3 \) also share one (not necessarily the same) player.
- This pattern continues such that team \( T_n \) and team \( T_1 \) also share exactly one player.
- Visualize the teams in a circular arrangement. The sequence of shared players forms a continuous loop around the circle of teams.
- Consider the overall distribution: if each team of \( n \) teams shares players with its neighbor, then each team initially contributes all \( k \) players, but each team is overlapping with its neighboring teams by one player per connection.
- To find unique players, subtract the overlapping players: since there are \( n \) pairs of teams sharing a player, \( n \) players are overlapping and subtracted.
- Hence, total unique players across all teams can be calculated by subtracting the \( n \) overlaps from the total players initially counted: \[ (n \times k) - n = n(k-1) \]
Therefore, the total number of unique players in the tournament is \( n(k-1) \).
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