Problem 62 - Algebra and Trigonometry

Question

On a basketball team of 12 players, 2 play only center, 3 play only guard, and the rest play forward (5 players on a team: 2 forwards, 2 guards, and 1 center). How many different teams are possible, assuming that it is not possible to distinguish a left guard from a right guard or a left forward from a right forward?

Step-by-Step Solution

Answer

126 different teams

Step 1: Identify players for each position

There are 12 players in total: 2 centers, 3 guards, and the rest (7 players) are forwards.

Step 2: Calculate the combinations for centers

Since only 1 center is needed and there are 2 available centers, the number of ways to select 1 center from 2 is \( \binom{2}{1} = 2 \).

Step 3: Calculate the combinations for guards

Since 2 guards are needed and there are 3 available, the number of ways to select 2 guards from 3 is \( \binom{3}{2} = 3 \).

Step 4: Calculate the combinations for forwards

Since 2 forwards are needed and there are 7 available, the number of ways to select 2 forwards from 7 is \( \binom{7}{2} = 21 \).

Step 5: Multiply the combinations

The total number of different teams is the product of the combinations of centers, guards, and forwards: \( 2 \times 3 \times 21 = 126 \).

Key Concepts

combinations

Combinations are a key concept in combinatorics. They are used to determine how many ways you can choose items from a group, without caring about the order of selection. For example, consider you have 3 fruits: an apple, a banana, and a cherry. If you want to choose 2 out of these 3 fruits, you can have the following combinations: apple and banana, apple and cherry, or banana and cherry. The number of ways to select 2 items from 3 is calculated using the formula for combinations \(\binom{n}{k}\), where n is the total number of items, and k is the number of items to choose. Mathematically, this is represented as \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here, \(!\) denotes factorial, which means multiplying a sequence of descending natural numbers. Combinations ensure that the order of selecting items does not matter. In our basketball team example, this concept helps in choosing centers, guards, and forwards effectively.

binomial coefficient

The binomial coefficient, often denoted as \( \binom{n}{k} \), is a specific mathematical term within combinatorics. It is used to compute the number of ways to select k items out of n items, where the order of selection does not matter. The binomial coefficient can also be seen in the expanded form of a binomial expression, such as \( (a + b)^n \). The binomial coefficient directly relates to combinations. For instance, in selecting the forwards for the basketball team, the number of ways to choose 2 forwards out of 7 is given by the binomial coefficient \( \binom{7}{2} = 21 \). This coefficient is crucial in the calculation steps to find the total number of possible teams.

team selection

Team selection often involves choosing members for different positions, following specific rules or constraints. In our problem, we have constraints like having 2 forwards, 2 guards, and 1 center. The selection steps are broken down based on the specific number of players available for each position and the number required.

• First, you determine the available and required players for each position.
• Then, you calculate the possible combinations using the binomial coefficient for each group.

Once all combinations are determined, multiply them to get the total number of possible teams. For example, for our basketball team:

• Centers: \( \binom{2}{1} = 2 \)
• Guards: \( \binom{3}{2} = 3 \)
• Forwards: \( \binom{7}{2} = 21 \)

Multiplying these gives the total number of teams: \( 2 \times 3 \times 21 = 126 \).

basic probability

Basic probability often uses combinatorics to determine the likelihood of an event. Probability is calculated by considering all possible outcomes and the desired outcomes. The formula is as follows: \[ \text{Probability} = \frac{\text{Desired outcomes}}{\text{Total possible outcomes}} \] In the context of our basketball team, while the exercise focuses on combinations, understanding probability can further help to predict the likelihood of forming a particular team configuration. By mastering combinatorial techniques like calculating combinations and using the binomial coefficient, you can seamlessly transition to solving more complex probability problems.