In a darts tournament two groups of 16 players each are drawn against each other and the winners go forward to the next round. Two players fail to turn up and their opponents are given an automatic win. What is the total number of matches that will have to be played to a winner?
A 29
B 30
C 31
D 32
Please explain. Much thanks
With 16 players , number of total games
= C(16,2) = 120
with only 14 players, number of total games
= C(14,2) = 91
120-91= 29
The wording of this question is not clear.
In a typical "knock-out" tournament, once you lose a game you are done
e.g. the current Australian Open
But you seem to suggest that every team plays every other team, (like
the typical "shake-hands" problem. That is why I used the C(n,r) approach
But then there would be just 91 games with the person with most wins wins.
In a 16 player tournament , there would be 8 games in round 1
4 games in round2, 2 games and finally the championship game.
Total number of games = 8+4+2+1 = 15
Such tournaments usually have number of entries that are powers of 2,
e.g. Australia has 128 in the Men's , 128 = 2^7
The ball is in your court.
Well, if we have two groups of 16 players each, that gives us a total of 32 players. Since two players didn't show up, we subtract them from the total, leaving us with 30 players who will actually play.
In each match, there will be one winner who goes forward to the next round. So if we have 30 players, we need 29 matches to determine all the winners (since the last match will determine the final winner).
So the correct answer is B) 30. But hey, who's counting? I'm just here for the laughs!
To find the total number of matches that need to be played in a darts tournament, we need to consider the number of matches in each round.
In the first round, there will be a total of 16 matches in each group. Since two players fail to turn up, their opponents receive an automatic win. Therefore, in each group, 14 matches will be played.
In the second round, the winners from each group will go forward. Thus, there will be 14 matches in total.
To determine the total number of matches, we add the matches from both rounds: 14 + 14 = 28.
However, we also need to account for the final match where the winner will be determined. So, we add one more match to the total: 28 + 1 = 29.
Therefore, the total number of matches that need to be played to get a winner in this darts tournament is 29.
Hence, the correct answer is option A) 29.
To find the total number of matches that will have to be played to determine a winner in the given darts tournament scenario, we need to consider the number of matches played in each round.
First, we start with two groups of 16 players each, which means a total of 32 players. In the initial round, all 32 players will compete against each other. Since two players fail to turn up, their opponents are given an automatic win. This reduces the number of matches in the first round from 32 to 30.
In each match, one player wins and advances to the next round, while the other player is eliminated. So, after the first round, we will have 30 winners remaining.
For the second round, these 30 winners will be drawn against each other. Each match will eliminate one player, and the number of remaining players will be halved. Thus, after the second round, we will have 15 winners.
In the third round, the 15 winners will be drawn against each other again, resulting in 7 matches. After this round, we will have 7 winners remaining.
Finally, in the fourth and last round, these 7 winners will compete against each other, resulting in 3 matches. After this round, we will have 3 winners.
To determine the tournament's overall winner, we need one final match between these 3 winners.
Adding up the number of matches played in each round:
First round: 30 matches
Second round: 15 matches
Third round: 7 matches
Fourth round: 3 matches
Final match: 1 match
So, the total number of matches that will have to be played to determine the winner is 30 + 15 + 7 + 3 + 1 = 56.
However, since the options provided for the answers are different, there might be some ambiguity or additional information missing from the question.