Problem 20 - Algebra for College Students

Question

In a baseball league of nine teams, how many games are needed to complete the schedule if each team plays 12 games with each other team? 432

Step-by-Step Solution

Answer

432 games are needed.

Step 1: Identify the Number of Teams

In this exercise, we have a total of 9 teams in the baseball league. This is the foundational data necessary for calculating the total number of games played.

Step 2: Determine Games per Team Pair

Each team in the league plays 12 games against each other team. This number ensures that each pairing of teams competes 12 times.

Step 3: Calculate the Total Number of Team Pairs

To find out how many different matchups there are, we need to use the combination formula to find the number of ways to choose 2 teams out of 9. This is calculated as follows:\[\binom{9}{2} = \frac{9 \times 8}{2} = 36\]This means there are 36 unique team pairs.

Step 4: Compute Total Games

Now, multiply the number of unique team pairs by the number of games each pair plays:\[\text{Total Games} = 36 \text{ pairs} \times 12 \text{ games per pair} = 432\]

Step 5: Conclusion: Calculate the Final Total

We now know there are a total of 432 games needed to complete the schedule for the league, ensuring that each team faces every other team 12 times.

Key Concepts

Combination Formula

The combination formula is an essential tool in combinatorial mathematics, especially when determining how many unique pairs or groups can be formed from a larger set. In mathematical terms, when selecting 2 teams from a group of 9, we can use the combination formula. This is represented as \(\binom{n}{k}\), where \(n\) is the total number of items (teams in this case), and \(k\) is the number of items to select. For example:

• \(n = 9\): Total teams.
• \(k = 2\): Teams per match.

To calculate, the formula is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]For our example, \(\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36\). This shows there are 36 unique ways to select 2 teams from 9, making it evident how the league's matchups are determined.

Team Pairing

Team pairing is crucial in organizing tournaments or leagues. It determines the opponents each team will face, and in this exercise, involves selecting two teams to compete against each other. For a league with multiple teams like a baseball league:

• Each team needs to play against every other team.
• Consistent matchups are required to ensure fairness.

By using the combination concept, we efficiently calculate the number of unique pairings. This not only helps in ensuring each team gets equal play but also aids in logistics for scheduling matches. Proper pairing guarantees that every pair gets equal opportunity, which is vital for competitive balance.

Game Scheduling

Scheduling games in a sports league involves planning when and where each match will occur. It's a complex task that must consider team availability and venue logistics while ensuring every team plays the required number of games. In this specific example:

• There are 36 unique team pairs, calculated using the combination formula.
• Each pair has to play 12 games, making a total of 432 games.

The schedule must manage these games across available dates and locations while accounting for additional factors like travel distances and rest days for players. Effective game scheduling balances the needs of the teams, players, and league organizers, fostering a smoothly-run tournament.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, which plays a significant role in combinatorial mathematics. In this context, it's fundamental to topics like the combination formula and is particularly impactful in solving problems such as team pairings and game scheduling:

• Involves techniques for counting combinations and permutations.
• Offers structured methodologies for problem-solving in finite systems.

The use of discrete mathematics here ensures precise calculation of game schedules and matchups, enabling leagues to function effectively. By structuring and solving arrangements of teams and games, discrete mathematics becomes an indispensable tool in sports management and beyond.