Permutations matches in basketball league

In summary, there are 81 total games for one team in the North Central Division to be scheduled. Using the formula for permutations, there are \frac{81!}{4!^4 \times 11!^3 \times 16!^2} possible orders for these games to be played. However, this calculation may be difficult without a calculator. It is also important to consider that these games are independent of each other, so the number of choices for each game is not affected by previous games.
  • #1
master cherundo
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0

Homework Statement


Suppose that a basketball league has 32 teams, split into two conferences of 16 teams each. Each conference is split into three divisions. Suppose that the North Central Division has five teams. Each of the teams in the North Central Division plays four games agains each of the other teams in this division, three games against each of the 11 remaining teams in the conference, and two games against each of the 16 teams in the other conference. In how many different orders can the games of one of the teams in the North Central DIvision be scheduled?



Homework Equations


[tex]\frac{n!}{n_1n_2 \cdots n_k}[/tex]


The Attempt at a Solution


Well, I think it would be [tex]\frac{81!}{4!^4 \times 11!^3 \times 16!^2}[/tex]
But, it is a hard stuff to compute this if we do not use calculator. And, I believe it has implementation with how to distribute [tex]82[/tex] objects into [tex]n[/tex] boxes. Unfortunately, I do not know how to find [tex]n[/tex].
 
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  • #2
I don't think you are understanding what a permutation is exactly. The definition of a permutation is:

[tex]_nP_k = \frac{n!}{(n-k)!}[/tex]

The reason for this is because, in the beginning, you have n objects from which to choose. For the second object, you have n-1 choices. Third object, you have n-2 choices. Kth object, you have n-k+1 choices. This multiplication is equivalent to the permutation definition above.

In your case, how many total games are there? Are the independent of each other--that is, does playing one game affect whether you can play any of the other games? Think about how many total choices you have for the first game, how many total choices you have for the second game, etc.
 
  • #3


I would approach this problem by first understanding the concept of permutations and the formula for calculating them. In this case, we have 81 games to be scheduled for one team in the North Central Division. Using the formula for permutations, we have 81! possible ways to arrange these games. However, we must also consider the fact that the team will play four games against each of the other four teams in their division, three games against each of the 11 remaining teams in the conference, and two games against each of the 16 teams in the other conference. This means that we must divide by the number of ways these games can be arranged within each category.

For the games within the division, we have 5 teams and 4 games each, so we have 5!^4 ways to arrange these games. For the games within the conference, we have 11 remaining teams and 3 games each, so we have 11!^3 ways to arrange these games. And for the games in the other conference, we have 16 teams and 2 games each, so we have 16!^2 ways to arrange these games.

Therefore, the total number of possible schedules for one team in the North Central Division would be:

\frac{81!}{5!^4 \times 11!^3 \times 16!^2} = 4.331 \times 10^{74} possible schedules.

This is a very large number, but it is important to consider all the different factors and possibilities when scheduling games in a basketball league. As a scientist, it is important to carefully consider all the variables and use mathematical formulas to find the most accurate and efficient solution.
 

1. What are permutations matches in a basketball league?

Permutations matches in a basketball league refer to the different ways in which teams can play against each other within a league. It is a way to determine the schedule and matchups for each team throughout the season.

2. How are permutations matches determined in a basketball league?

Permutations matches are determined through a process of randomization or seeding. This ensures that each team has a fair chance of playing against all other teams in the league.

3. What is the purpose of having permutations matches in a basketball league?

The purpose of having permutations matches is to create a balanced and fair schedule for all teams in the league. It also adds excitement and unpredictability to the season as teams may face different opponents each time they play.

4. Can permutations matches affect a team's ranking in the basketball league?

Yes, permutations matches can affect a team's ranking in the league. The outcome of each match can impact a team's overall record and standing within the league.

5. Is there a limit to the number of permutations matches in a basketball league?

There is no specific limit to the number of permutations matches in a basketball league. It depends on the number of teams in the league and the length of the season. However, most leagues aim to have each team play against every other team at least once.

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