Suppose that n tennis players want to play an elimination tournament. That means: they pair up, at random, for each round; if the number of players before the round begins is odd, one of them, chosen at random, sits out that round. The winners of each round, and the odd one who sat it out (if there was an odd one), play in the next round, till, finally, there is only one winner, the champion. What is the total number of matches to be played altogether, in all the rounds of the tournament?
n-1
To find the total number of matches to be played in the elimination tournament, we can break it down into rounds.
In the first round, there will be n/2 matches played since each pair of players will play against each other. If n is odd, one player will sit out, so there will be (n-1)/2 matches.
In the second round, the winners from the first round will play. The number of players remaining will be half of the previous round, so there will be (n/2)/2 = n/4 matches.
Following this pattern, in each subsequent round, the number of matches will be halved until we have only one winner.
Therefore, the total number of matches can be found by summing up the number of matches in each round:
Number of matches in round 1: (n-1)/2
Number of matches in round 2: (n/4)
Number of matches in round 3: (n/8)
...
Number of matches in the final round: 1
To find the total, we can set up the following equation:
Total number of matches = (n-1)/2 + n/4 + n/8 + ... + 1
This is a geometric series, where the common ratio is 1/2. The sum of a geometric series can be found using the formula:
S = a(1 - r^n) / (1 - r)
Where "a" is the first term, "r" is the common ratio, and "n" is the number of terms.
In this case, the first term is (n-1)/2, the common ratio is 1/2, and the number of terms is log2(n).
Plugging in the values into the formula, we get:
Total number of matches = ((n-1)/2) * (1 - (1/2)^log2(n)) / (1 - 1/2)
Simplifying further, we can write it as:
Total number of matches = (n-1) * (1 - 1/n) / (1 - 1/2)
= 2(n-1) * (1 - 1/n)
= 2(n-1 - 1 + 1/n)
= 2(n - 1 + 1/n)
Thus, the total number of matches to be played altogether in all the rounds of the tournament is 2(n - 1 + 1/n) matches.
To find the total number of matches to be played in the tournament, we need to determine the number of matches in each round and the number of rounds in the tournament.
Let's start by understanding the number of matches in each round. In each round, every player who is participating will play one match against another player. If there are 'n' players, we will have 'n/2' matches in that round. For example, if there are 8 players, there will be 4 matches in the first round.
Now, let's calculate the number of rounds in the tournament. We need to keep playing rounds until there is only one winner left. In each round, the number of players gets halved (except for the cases when there is an odd number of players). The tournament will end when we have one player left. So, the number of rounds will be equal to the logarithm (base 2) of 'n' rounded up to the nearest integer. For example, if there are 8 players, we will have 3 rounds.
Therefore, the total number of matches in all the rounds can be calculated by summing the number of matches in each round. Since each round has 'n/2' matches, and there are a total of 'log2(n)' rounds, the formula to calculate the number of matches is:
Total matches = (n/2) + (n/4) + (n/8) + ... + 1
This is a geometric series with a common ratio of 1/2. To find the sum of a geometric series, we can use the formula:
Sum = (first term * (1 - common ratio^n)) / (1 - common ratio)
In this case, the first term is 'n/2', the common ratio is 1/2, and the total number of terms is 'log2(n)'. Plugging these values into the formula, we get:
Total matches = (n/2 * (1 - (1/2)^log2(n))) / (1 - 1/2)
Simplifying this expression will give us the total number of matches to be played in the tournament.