10 teams enter a soccer tournament. In the first round teams paired in 5 pairs to play with each other.
Assume that now in each match one of the teams marked as "HOME" and the other "AWAY" and first 5 games need to run one after another. How many different pairings are possible to create?
To determine the number of different pairings that can be created, we'll use the concept of permutations.
In the first round, there are 10 teams and they are paired in 5 pairs. Since order matters, we can think of this problem as arranging the teams in a specific order.
We can start by picking one team to be the "HOME" team for the first match. There are 10 teams to choose from, so we have 10 options.
Once we have chosen the "HOME" team for the first match, we can now choose one team to be its opponent, the "AWAY" team. However, since the teams have already been paired up, there are only 9 teams remaining to choose from.
For the second match, we have 8 teams left to choose from for the "HOME" team, as one team has already been chosen to play. Similarly, there will only be 7 teams remaining to choose from for the "AWAY" team.
We continue this process for each subsequent match until we reach the fifth match. For each match, the number of options for choosing the "HOME" team decreases by 1, while the number of options for choosing the "AWAY" team decreases by 1 as well.
To calculate the total number of pairings, we multiply the number of options for each match together:
10 * 9 * 8 * 7 * 6 = 30,240
Therefore, there are 30,240 different pairings that can be created for the first round of the soccer tournament.