Question

A basketball team has 12 players. However, only five players play at any given time during a game. In how may ways can the coach choose the five players? To be more realistic, the five players playing a game normally consist of two guards, two forwards, and one center. If there are five guards, four forwards, and three centers on the team, in how many ways can the coach choose two guards, two forwards, and one center? What if one of the centers is equally skilled at playing forward?

Short answer

There are 792 ways to choose any 5 players, 180 with position constraints, and 320 considering player flexibility.

Step-by-step solution

Understanding the Basic Combination

First, let's determine how many ways the coach can choose 5 players from 12. This is a classic combination problem, denoted as \( C(n, k) \), where \( n \) is the total number of players, and \( k \) is the number of players to choose. This is calculated as \( C(12, 5) = \frac{12!}{5!(12-5)!} \).

Calculate Total Combinations

Apply the formula to calculate \( C(12, 5) \): \[ C(12, 5) = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 \]. So there are 792 ways to choose any 5 players from 12.

Determine Guard Selection Combinations

Now, let's select 2 guards out of 5. Use the combination formula: \( C(5, 2) = \frac{5!}{2!(5-2)!} \). Calculating this gives: \[ C(5, 2) = \frac{5 \times 4}{2 \times 1} = 10 \].

Determine Forward Selection Combinations

Next, select 2 forwards out of 4 using the formula: \( C(4, 2) = \frac{4!}{2!(4-2)!} \). Calculating this gives: \[ C(4, 2) = \frac{4 \times 3}{2 \times 1} = 6 \].

Determine Center Selection Combinations

Select 1 center out of 3 using the formula: \( C(3, 1) = \frac{3!}{1!(3-1)!} \). Calculating this gives: \[ C(3, 1) = 3 \].

Combine Selections for Total Combinations

Now, multiply the combinations for selecting guards, forwards, and centers: \( 10 \times 6 \times 3 = 180 \). This means there are 180 ways to choose the players given the position constraints.

Adjust for Skilled Center

If one center is equally skilled at playing forward, this player introduces additional flexibility. Count him either as a center or a forward and calculate separately for each scenario, then sum them.

Calculate Adjusted Combinations

First, consider him as a center: \( 10 \times 6 \times 2 = 120 \). Then, consider him as a forward. Choose 2 from now 5 forwards: \( C(5, 2) = 10 \). Calculate: \( 10 \times 10 \times 2 = 200 \). Add the two scenarios: \( 120 + 200 = 320 \).

Final Answer

With the multi-skilled player, there are a total of 320 ways to choose the players.

Key concepts

Combinations

Combinations are a fundamental concept in combinatorics, a branch of mathematics focusing on the study of finite or countable discrete structures. In problems involving combinations, the order of elements does not matter—what is important is the selection of a certain number of elements from a set.

When we talk about combinations, we often use a formula to calculate the number of ways to select items. This is represented by \( C(n, k) \), where \( n \) is the total number of items and \( k \) is the number of items to choose. The formula is:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

This formula accounts for choosing \( k \) items out of \( n \) without regard to the order of selection. "\( ! \)" is a factorial, which means multiplying a series of descending natural numbers. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Using combinations can simplify complex problems, such as selecting players for a basketball team or decision-making in group settings.

Basketball Team Selection

Basketball team selection often involves making strategic choices about the composition of the players on the court. Given certain constraints, such as specific numbers of players for each position, combination techniques can be used to calculate all possible arrangements.

Consider selecting a basketball lineup from a team. If you need to choose 5 players out of 12, this problem can be solved by computing \( C(12, 5) \), which results in 792 possible ways. However, coaches usually have specific needs, such as choosing a certain number of guards, forwards, and centers.

• Selecting 2 guards from 5 results in \( 10 \) combinations (\( C(5, 2) = 10 \)).
• Selecting 2 forwards from 4 results in \( 6 \) combinations (\( C(4, 2) = 6 \)).
• Selecting 1 center from 3 results in \( 3 \) combinations (\( C(3, 1) = 3 \)).

For these specific position constraints, there are 180 ways to fill the lineup by multiplying the combinations for each position: \( 10 \times 6 \times 3 = 180 \).

Such methodical approaches ensure that all strategic factors are considered, providing coaches with a comprehensive set of options to choose the best combination of players.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with discrete elements that uses algebra and arithmetic. It is integral in various computing scenarios and mathematical theories.

Several areas fall under discrete mathematics, including combinatorics, graph theory, and logic. Each of these areas provides tools for analyzing problems where distinct or countable values are essential.

In our basketball team selection problem, discrete mathematics plays a role through combinatorics, which systematizes the calculation of possible player arrangements. Calculating combinations and considering constraints such as player positions are applications of discrete mathematics principles.

• Combinatorics helps find ways to arrange and select groups of items.
• Systems like logical reasoning are used to deduce solutions systematically and accurately.

Discrete mathematics provides the backbone for theoretical and practical applications, enabling data analysis, problem-solving, and decision-making across diverse fields.