To seed a tournament means to select which teams will play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15.† This exercise is based on various types of elimination tournaments. (Leave each answer as a formula. Do not enter your answers using the notation
P(n, r)
or
C(n, r).)How many different seedings of a 16-team tournament are possible?

To determine how many different seedings of a 16-team tournament are possible, we need to find the number of ways to assign the 16 teams to the specific seeding positions based on their preliminary ranking.

### Step-by-step approach:

1. There are 16 teams in total, and we need to assign them to 16 specific seeding positions (1 through 16).

2. The first position (seed 1) can be filled by any of the 16 teams.

3. The second position (seed 2) can be filled by any of the remaining 15 teams.

4. The third position (seed 3) can be filled by any of the remaining 14 teams, and so on.

5. This process continues until all teams are assigned to a position.

This means the total number of different seedings is the number of ways to assign 16 teams to 16 positions, which is simply the number of permutations of 16 teams.

The number of permutations of \(n\) teams is given by:

\[
n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1
\]

In this case, \(n = 16\), so the number of different seedings is:

\[
16! = 16 \times 15 \times 14 \times \cdots \times 1
\]

Thus, the total number of different seedings for a 16-team tournament is \( 16! \).

If you want to explore more, you can calculate \(16!\) to find the exact number.

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Do you want further details or clarifications?

Here are 5 related questions to explore:
1. How many different seedings are possible for an 8-team tournament?
2. What is the formula for the number of seedings for an \(n\)-team tournament?
3. How does the formula change for a double-elimination tournament?
4. What is the probability of a specific team getting the top seed in a random draw?
5. How does the seeding system affect the outcome of the tournament?

**Tip:** When dealing with permutations, remember that \(n!\) grows extremely fast, so even small values of \(n\) can lead to large numbers.

To seed a tournament means to select which teams will play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15.† This exercise is based on various types of elimination tournaments. How many different seedings of a 16-team tournament are possible?

Learn how to calculate the number of possible seedings for a 16-team tournament. This problem involves combinatorics and the concept of permutations, and it explains how factorial calculations determine the number of ways to assign teams to positions based on their rankings. Suitable for students in Grades 10-12.

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