Problem 52 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A company car that has a seating capacity of six is to be used by six employees who have formed a car pool. If only four of these employees can drive, how many possible seating arrangements are there for the group?

Step-by-Step Solution

Answer

There are 480 possible seating arrangements for the group of six employees in the car pool.

Step 1: Identify the different cases

We first need to identify the different cases in this problem. In this case, we have 3 major aspects to consider: 1. The driver's seat: There are four employees who can occupy this position. 2. The remaining five seats: The remaining five employees can occupy these seats. 3. The total number of seating arrangements: We need to calculate the different ways these employees can sit in the car. For clarity, let us assume the employees capable of driving are A, B, C, and D while the non-drivers are E and F.

Step 2: Calculate the number of ways to choose the driver

There are four employees who can drive, therefore, there are four ways to choose the driver.

Step 3: Calculate the number of ways the remaining employees can sit

We have five remaining seats since one has been occupied by the driver. Let's assume the driver is A. We now have 5 other employees (B, C, D, E, and F) to fill these seats. We can find this by calculating the permutation of 5 elements: \(P(5) = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\) There are 120 ways the remaining employees can sit after the driver's seat is occupied.

Step 4: Calculate the total number of seating arrangements

Considering both choosing the driver and the remaining seating arrangements, we have: Total seating arrangements = (Number of ways to choose the driver) × (Number of ways the remaining employees can sit) Total seating arrangements = 4 × 120 Total seating arrangements = 480 There are 480 possible seating arrangements for the group of six employees in the car pool.

Key Concepts

Seating Arrangements

Understanding seating arrangements problems involves both logic and mathematics, especially when dealing with permutations. Think of a seating arrangement as a method to organize objects (in this case, people) in a specific order. When it comes to seating people in a car, we need to consider who can sit where. For instance, not everyone may be able to drive, which limits who can sit in the driver's seat.

In our example, with a car that has six seats and four potential drivers, the situation becomes a classic permutation problem because each driver changes the possible arrangements of the remaining passengers. It's essential to consider every possible scenario. For example, if one particular employee is always the driver, other employees have fewer options where to sit. As this exercise shows, permutation problems often require breaking the problem down into different cases to simplify it and ensure all options are accounted for.

Factorial

The factorial, denoted as an exclamation point (!), is fundamental to combinatorics, the branch of mathematics that deals with counting combinations and permutations. The factorial of a non-negative integer n, expressed as n!, is the product of all positive integers less than or equal to n. For example, 5! is 5 × 4 × 3 × 2 × 1, which equals 120.

In our carpool scenario, calculating factorials helps determine the total number of seat arrangements for passengers after choosing the driver. A simple exercise improvement advice is to remember that 0! is always 1, which is a principle that sometimes puzzles students but is crucial when dealing with empty sets in permutations.

Permutations

The concept of permutations addresses the arrangement of items where the order does matter. In mathematical terms, a permutation is an ordered combination. Specifically, for a set with a certain number of elements (n), the number of distinct permutations is given by n!. So, when we talk about seating arrangements, we're referring to the various permutations of people in different seats.

Let's apply this to our car pool example. After choosing the driver, who has 4 possibilities, we deal with the permutations of the remaining five employees for the other seats, leading us to calculate 5! or 120 scenarios. Multiplying the permutations of driver choices by the passengers' permutations, which means 4 × 120, we find there are 480 unique ways to arrange the six employees in the car.