Problem 35 - Algebra for College Students

Question

Ten basketball players are gomg to be divided into two teams of five players each for a game. In how many ways can this be done?

Step-by-Step Solution

Answer

There are 126 ways to form two teams.

Step 1: Understand the total group size

We have a total of ten players that need to be divided into two teams.

Step 2: Choose players for the first team

The choice of players for the first team entirely determines both teams. Thus, we need to choose 5 players from a group of 10.

Step 3: Use Combination Formula

To calculate the number of ways to choose 5 players out of 10, we use the combination formula: \[\binom{10}{5} = \frac{10!}{5!(10-5)!}\]

Step 4: Calculate Factorials

Calculate the factorial of relevant numbers:- \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)- \(5! = 5 \times 4 \times 3 \times 2 \times 1\)- \(5! = 5 \times 4 \times 3 \times 2 \times 1\)

Step 5: Substitute and Simplify

Substitute the factorials into the combination formula:\[\binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = \frac{30240}{120} = 252\]

Step 6: Account for identical teams

Since choosing team A is the same as choosing team B and vice versa, divide by 2 to avoid double counting:\[\frac{252}{2} = 126\]

Step 7: Conclusion

Thus, there are 126 different ways to divide the ten basketball players into two teams of five players each.

Key Concepts

Understanding the Combination Formula

The combination formula is a key concept in combinatorics. It allows us to calculate the number of ways to choose a subset of items from a larger set without considering the order of selection. This is particularly useful when forming groups or teams where the arrangement doesn't matter. The formula for combinations, often denoted as \( \binom{n}{r} \), is given by:

• \( \binom{n}{r} = \frac{n!}{r! (n-r)!} \)

Here, \( n \) is the total number of items you can choose from, and \( r \) is the number of items you want to select. This formula simplifies our calculations by directly plugging in factorial values. For example, selecting 5 players from 10 involves calculating \( \binom{10}{5} \). This formula helps in determining that there are 252 ways to form one of the teams.

Exploring Factorials

Factorials are an essential part of combinatorial formulas like combinations and permutations. A factorial, denoted by \( n! \), is the product of all positive integers up to \( n \). For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \)

Factorials grow very quickly, which is why they're often used in problems involving large numbers. In combination calculations, they help quantify the possible sets or orderings and simplify the computation by cancellation of terms. They are crucial for dividing larger calculations into manageable parts, aiding in the simplification seen in \( \binom{10}{5} = \frac{10!}{5!5!} \).

Distinguishing Permutations

Permutations are similar to combinations but with one crucial difference: permutations consider the arrangement of items, making the order important. While combinations like \( \binom{n}{r} \) involve forming groups regardless of sequence, permutations use a different formula to account for different orders of selection. The permutation formula is given by:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This calculates the number of ways to arrange \( r \) items out of \( n \) distinct items. If we were to decide not just the teams but also the lineup order within the team, we would use permutations instead. However, since in our context, only the group formation matters, not the order, combinations are the appropriate choice.

Applying Mathematical Reasoning

Mathematical reasoning involves logical thinking to solve problems like team formation efficiently. To divide 10 basketball players into two teams, the choice of players for the first team automatically decides the second team. This realization reduces the problem to a single decision, simplifying calculations greatly.
Using the formula for combinations, we consider only the selection of the first team and then adjust by recognizing teams A and B are identical if their players simply switched sides, thus dividing our result by two.
This approach highlights the importance of considering all factors in a problem. Simplifying assumptions and recognizing patterns are key aspects of mathematical reasoning to achieve elegant solutions. Hence by acknowledging the symmetry of teams, we reach the conclusion of having 126 distinct team configurations.