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The 16 players in a chess tournament must each play every other player

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Bunuel
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The 16 players in a chess tournament must each play every other player [#permalink] New post   27 Jan 2022, 07:15
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The 16 players in a chess tournament must each play every other player in the first round of the tournament. Only half of the first 16 players will play in the second round, each player playing every other player, and only half of them will play in the third round, a pattern that will continue until one winner is left. How many chess games will be played in the tournament?

(A) 16
(B) 30
(C) 72
(D) 120
(E) 155
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E
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BrentGMATPrepNow
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Re: The 16 players in a chess tournament must each play every other player [#permalink] New post   27 Jan 2022, 07:51
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Bunuel wrote:
The 16 players in a chess tournament must each play every other player in the first round of the tournament. Only half of the first 16 players will play in the second round, each player playing every other player, and only half of them will play in the third round, a pattern that will continue until one winner is left. How many chess games will be played in the tournament?

(A) 16
(B) 30
(C) 72
(D) 120
(E) 155


Let's start by calculating the total number of games played by the original 16 players.
In other words, in how many ways can we select two players two play a game of chess?
Since the order in which we select the two players does not matter, we can use combinations.
We can select 2 players from 16 players in 16C2 ways.
16C2 = (16)(15)/(2)(1) = 120

Once the first round of play eliminates half the players, we're left with 8 players.
Important: Since there are still more games to be played, we know the correct answer will be greater than 120....
....Answer: E
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Kimberly77
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Re: The 16 players in a chess tournament must each play every other player [#permalink] New post   04 Dec 2022, 03:54
BrentGMATPrepNow wrote:
Bunuel wrote:
The 16 players in a chess tournament must each play every other player in the first round of the tournament. Only half of the first 16 players will play in the second round, each player playing every other player, and only half of them will play in the third round, a pattern that will continue until one winner is left. How many chess games will be played in the tournament?

(A) 16
(B) 30
(C) 72
(D) 120
(E) 155


Let's start by calculating the total number of games played by the original 16 players.
In other words, in how many ways can we select two players two play a game of chess?
Since the order in which we select the two players does not matter, we can use combinations.
We can select 2 players from 16 players in 16C2 ways.
16C2 = (16)(15)/(2)(1) = 120

Once the first round of play eliminates half the players, we're left with 8 players.
Important: Since there are still more games to be played, we know the correct answer will be greater than 120....
....Answer: E


Hi BrentGMATPrepNow, I calculate it by listings as below and not sure why is not right?
Player 16 against Player 15 ... 1st game
Player 15 against Player 14 ... 2nd game
Player 14 against Player 13 ... 3rd game
....
Player 2 against Player 1 ... 15th game

So there are 15 chess games will be played in the tournament.
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BrentGMATPrepNow
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Re: The 16 players in a chess tournament must each play every other player [#permalink] New post   04 Dec 2022, 08:24
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Kimberly77 wrote:

Hi BrentGMATPrepNow, I calculate it by listings as below and not sure why is not right?
Player 16 against Player 15 ... 1st game
Player 15 against Player 14 ... 2nd game
Player 14 against Player 13 ... 3rd game
....
Player 2 against Player 1 ... 15th game

So there are 15 chess games will be played in the tournament.


The question tells us that The 16 players in a chess tournament [i]must each play every other player in the first round of the tournament[/i]
And your solution, Player 16 plays only one game (against Player 15)
In actuality (to follow the conditions set out in the question), Player 16 must play Player 15, Player 14, Player 13, Player 12, . . . . Player 3, Player 2 and Player 1.
Similarly, Player 15 must play Player 16, Player 14, Player 13, Player 12, . . . . Player 3, Player 2 and Player 1.
Similarly, Player 14 must play Player 16, Player 15, Player 13, Player 12, . . . . Player 3, Player 2 and Player 1.
As you can see, I've already listed more than 15 games.
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Kimberly77
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Re: The 16 players in a chess tournament must each play every other player [#permalink] New post   04 Dec 2022, 12:50
BrentGMATPrepNow wrote:
Kimberly77 wrote:

Hi BrentGMATPrepNow, I calculate it by listings as below and not sure why is not right?
Player 16 against Player 15 ... 1st game
Player 15 against Player 14 ... 2nd game
Player 14 against Player 13 ... 3rd game
....
Player 2 against Player 1 ... 15th game

So there are 15 chess games will be played in the tournament.


The question tells us that The 16 players in a chess tournament [i]must each play every other player in the first round of the tournament[/i]
And your solution, Player 16 plays only one game (against Player 15)
In actuality (to follow the conditions set out in the question), Player 16 must play Player 15, Player 14, Player 13, Player 12, . . . . Player 3, Player 2 and Player 1.
Similarly, Player 15 must play Player 16, Player 14, Player 13, Player 12, . . . . Player 3, Player 2 and Player 1.
Similarly, Player 14 must play Player 16, Player 15, Player 13, Player 12, . . . . Player 3, Player 2 and Player 1.
As you can see, I've already listed more than 15 games.


Thanks BrentGMATPrepNow, Get it and see why now. :please:
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Waqas.Baig
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Re: The 16 players in a chess tournament must each play every other player [#permalink] New post   01 Feb 2024, 13:21
Hi All, kindly evaluate my approach: based on Brent concept, 4stages will be there as with each stage player is getting halved. So in stage 16player will play against 15others total no of games will be 16*15/2 (divided by 2 as duplicacy is there player 1 playing player 2 and player 2playing against player 1 counted twice). similarly, stage2: 8*7/2 stage 3: 4*3/2 and stage 4: 2*1/2 =155
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Re: The 16 players in a chess tournament must each play every other player [#permalink]
01 Feb 2024, 13:21
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