Problem 64 - Algebra and Trigonometry

Question

Solve the problem using the appropriate counting principle(s). Doubles Tennis From a group of ten male and ten female tennis players, two men and two women are to face each other in a men-versus-women doubles match. In how many different ways can this match be arranged?

Step-by-Step Solution

Answer

There are 2025 different ways to arrange the match.

Step 1: Choose the Men

First, we need to choose two men from the group of ten male players. The number of ways to choose 2 players from a set of 10 is given by the combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Therefore, for the men: \( \binom{10}{2} = \frac{10!}{2!(10-2)!} = 45 \).

Step 2: Choose the Women

Similarly, we need to choose two women from the group of ten female players. We use the same combination formula: \( \binom{10}{2} = \frac{10!}{2!(10-2)!} = 45 \) ways.

Step 3: Multiply the Independent Choices

The total number of different matches can be found by multiplying the number of ways to choose the men by the number of ways to choose the women. Thus, the total number of ways to arrange the match is \( 45 \times 45 = 2025 \).

Key Concepts

Counting Principles

Counting principles help us figure out how many different ways a set of events can happen. In this exercise, we are dealing with a scenario involving a selection problem. Here, we want to choose players for two teams in a tennis match, and counting principles guide us to understand how many combinations are possible. They serve as the foundation for determining the total number of arrangements without actually having to list them all.

Two main types of counting principles often used are the **Addition Principle** and the **Multiplication Principle**.

• The **Addition Principle** applies when we have mutually exclusive situations - for example, choosing one option from either set A or set B.
• The **Multiplication Principle** is applied when we have to complete multiple independent tasks, like choosing team members.

In our tennis match problem, we use the Multiplication Principle; we count the choices for the men's team and multiply by the choices for the women's team to get the total possible arrangements.

Combination Formula

The combination formula is a mathematical way to determine how many ways we can choose a certain number of elements from a larger set without considering the order. This concept is especially important in our scenario because the order of picking team members doesn't matter, only the selection does, which is why we use combinations instead of permutations.

The formula for combinations is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] In this formula:

• \(n!\) indicates the factorial of \(n\). The factorial is the product of all positive integers up to \(n\).
• \(k\) stands for the number of elements to choose.
• The result \(\binom{n}{k}\) tells us the total number of ways to make these selections.

In our males and females tennis group, we calculate:

• For the men: \(\binom{10}{2} = 45\)
• For the women: \(\binom{10}{2} = 45\)

These calculations show how the combination formula aids in determining the possible arrangements for forming each team.

Doubles Tennis Match Arrangement

Arranging a doubles tennis match brings a practical application of combinatorics into a real-world scenario. The aim is to select two men and two women to compete against each other. In essence, we are organizing two teams from a larger group.

The challenge involves choosing two players from each gender group and ensuring that these choices reflect all possible combinations. By applying the combination formula as discussed in the previous sections:

• We first select 2 men from 10, calculating 45 combinations.
• We then select 2 women from 10, also resulting in 45 combinations.

Once both teams are chosen, we combine these independent choices. This is done using the Multiplication Principle to ascertain the complete set of possible doubles matches:\[45 \times 45 = 2025 \]This means there are 2025 unique ways to organize such doubles matches, illustrating the versatility of counting principles in organizing events.