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  • A chess tournament has 10 players. In how many different ways can the players be paired for the first round if each pairing consists of one player with a higher rating and one player with a lower r

A chess tournament has 10 players. In how many different ways can the players be paired for the first round if each pairing consists of one player with a higher rating and one player with a lower rating?

Option: 1

3,628,800

 


Option: 2

4,568,459

 


Option: 3

1,258,456

 


Option: 4

2,145,789


Answers (1)

best_answer

To calculate the number of different ways the players can be paired for the first round of a chess tournament, where each pairing consists of one player with a higher rating and one player with a lower rating, we can follow these steps:

We first need to select the players for each pairing. Since each pairing consists of one player with a higher rating and one player with a lower rating, we can pair the players in a way that the higher-rated player is always chosen first.

For each pairing, we need to select one player with a higher rating and one player with a lower rating. We have a total of 10 players, and we can choose one higher-rated player from the 10 , and then one lower-rated player from the remaining 9 players. Therefore, the number of ways to select one pairing is 10 \times 9=90.

Once the first pairing is selected, we move to the next pairing. For each subsequent pairing, the number of available higher-rated players decreases by 1 , and the number of available lower-rated players also decreases by 1 .

Therefore, for each subsequent pairing, we multiply the number of available higher-rated players by the number of available lower-rated players. In the second pairing, we have 8 higher-rated players and 7 lower-rated players to choose from. In the third pairing, we have 6 higher-rated players and 5 lower-rated players, and so on.

Continuing this process, we multiply the number of available higher-rated players and lower-rated players for each subsequent pairing until we have paired all the players.

Therefore, the total number of different ways the players can be paired for the first round, with each pairing consisting of one player with a higher rating and one player with a lower rating, is:

10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=3,628,800 .

Therefore, there are 3,628,800 different ways the players can be paired for the first round of the chess tournament, satisfying the condition that each pairing consists of one player with a higher rating and one player with a lower rating.

Posted by

Ajit Kumar Dubey

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