Question:
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:
\( T_1 \) and \( T_2 \),
\( T_2 \) and \( T_3 \),
...
\( T_{n-1} \) and \( T_n \),
\( T_n \) and \( T_1 \).
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:
\( T_1 \) and \( T_2 \),
\( T_2 \) and \( T_3 \),
...
\( T_{n-1} \) and \( T_n \),
\( T_n \) and \( T_1 \).
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
When counting elements with overlaps, subtract the repeated count due to shared elements.
Updated On: Aug 7, 2025
- \( k(n - 1) \)
- \( n(k - 2) \)
- \( k(n - 2) \)
- \( n(k - 1) \)
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The Correct Option is D
Solution and Explanation
Each team has \( k \) players, but adjacent teams share one player.
Since there are \( n \) such pairs (because \( T_n \) and \( T_1 \) also share one), total shared players = \( n \).
So, if we count all \( nk \) players initially, we are overcounting shared players.
Each shared player is counted twice, but is only one unique player.
Hence, total unique players:
\[
= nk - n = n(k - 1)
\]
\[
\boxed{n(k - 1)}
\]
\fbox{Final Answer: (D) \( n(k - 1) \)}
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