Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

STAFFING Student Painters, which specializes in painting the exterior of residential buildings, has five people available to be organized into two- person and three-person teams. a. In how many ways can a two-person team be formed? b. In how many ways can a three-person team be formed? c. In how many ways can the company organize the available people into either two-person teams or threeperson teams?

Step-by-Step Solution

Answer

There are 10 different ways to form a two-person team and 10 different ways to form a three-person team. So, there are a total of 20 different ways to organize the available people into either two-person teams or three-person teams.

Step 1: Forming a two-person team

We are given 5 people and need to form a team of 2. Using the combination formula: \[C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4}{2} = 10\] So there are 10 different ways to form a two-person team.

Step 2: Forming a three-person team

Similarly, for forming a three-person team, we substitute \(k = 3\) in the combination formula: \[C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4}{2} = 10\] So there are also 10 different ways to form a three-person team.

Step 3: Organizing into either two-person or three-person teams

For this part, we need to find the total number of ways to organize the people into either 2-person or 3-person teams. We have 10 ways to form a 2-person team and 10 ways to form a 3-person team, so the total ways to organize them is simply the sum of both: \[10 + 10 = 20\] Hence, there are 20 different ways to organize the available people into either two-person teams or three-person teams.

Key Concepts

Combination Formula

In the world of combinatorics, the combination formula is a crucial tool that allows us to calculate the number of ways to choose a subset of items from a larger pool without considering the order. This is different from permutations where the order does matter.
For any given set of objects, the combination formula is expressed mathematically as:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Here, \(n\) denotes the total number of items, \(k\) represents the number of items to be selected, and \(!\) stands for factorial, which means a product of all positive integers up to that number.
In simpler terms, when forming a team out of a group, using the combination formula helps you determine how many different ways you can select your team members, regardless of how they're arranged.

Team Formation

In team formation, you apply the combination formula to draw out an effective way of selecting team members from a larger group. Let's take an example of having 5 potential team members. You might want to form a team of 2 to tackle a specific task.

Two-Person Team

Using the combination formula, you calculate the number of ways by plugging in \(n=5\) and \(k=2\). This gives you:

• \( C(5, 2) = \frac{5!}{2!(5-2)!} = 10 \)

Therefore, there are 10 unique ways to form a two-person team.
On the other hand, if you decide to form a team of 3, the same formula can be adjusted by setting \(k=3\).

Three-Person Team

Plugging in the values we get:

• \( C(5, 3) = \frac{5!}{3!(5-3)!} = 10 \)

Thus, there are also 10 ways to form a three-person team. This illustrates how flexible and useful the combination formula is when you're in need of organizing groups into teams of various sizes.

Permutations and Combinations

Understanding permutations and combinations is essential in combinatorics, especially when organizing teams or selecting members from a group.

Permutations

Permutations are used when the order of selection matters. If arranging members in a lineup or assigning distinct roles is part of the task, permutations are the method to use. They typically result in larger numbers because of the specific arrangements considered.

Combinations

Combinations, conversely, disregard order and are ideal for team formation tasks where the grouping itself is the primary concern.

• For example, forming teams from available staff without specific roles is a classic combination problem.

Application in Exercise

In our exercise, the question involves forming teams without caring about the order in which members are selected. Therefore, it's combinations, not permutations, that apply. Summing up the possible team formations to consider both two-person and three-person groups gives us the total number of ways people can be organized, characterizing the understandability and simplicity behind combinations.