Problem 19 - Algebra for College Students

Question

In how many ways can Al, Bob, Carol, Dawn, and Ed be seated in a row of five chairs if \(A l\) is to be seated in the middle chair? 24

Step-by-Step Solution

Answer

There are 24 ways to seat them with Al in the middle.

Step 1: Examine the Problem Statement

We need to find the number of ways to arrange five people (Al, Bob, Carol, Dawn, and Ed) in a row when Al is fixed in the middle chair. This means that there are four remaining chairs for the rest of the individuals.

Step 2: Fix Al's Position

Since Al must be in the middle chair, he is already placed, reducing the problem to arranging the remaining four individuals (Bob, Carol, Dawn, and Ed) in the four chairs around him.

Step 3: Calculate Permutations for Remaining Seats

The four remaining people can be seated in the remaining four seats in different ways. This is a permutation problem for four distinct individuals, which can be calculated as the factorial of 4.

Step 4: Perform Calculation

Compute the factorial of 4, which is denoted as 4!. This is calculated as follows: \[4! = 4 \times 3 \times 2 \times 1 = 24\]

Step 5: Conclude the Solution

Since there are 24 ways to arrange the remaining four people, the total number of ways to arrange everyone with Al in the middle is 24.

Key Concepts

Factorial

The concept of a factorial is crucial when dealing with permutations and arrangement problems. Factorial, denoted by an exclamation mark \(n!\), represents the product of all positive integers up to a given number, \(n\). For example, the factorial of 4, written as \(4!\), is calculated by multiplying 4 by every positive integer less than it: \[4! = 4 \times 3 \times 2 \times 1 = 24\]
Factorials are a fundamental building block in combinatorics and are used to determine the number of ways to arrange a set of objects. They grow rapidly with increasing \(n\). For instance, \(5!\) equals \(120\), and \(6!\) equals \(720\). Understanding how to calculate factorials is essential for solving permutation-based problems effectively.

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combination of objects. It plays a significant role in solving problems related to permutations and arrangements, like the one involving Al, Bob, Carol, Dawn, and Ed seating arrangement.
Combinatorics mainly handles two kinds of problems:

• Permutations: These deal with the arrangement of objects in a specific order. For instance, if you're arranging four people in a row, you use factorials to find the number of permutations.
• Combinations: These involve selecting items from a group, where the order does not matter. While combinations are not directly used in our seating problem, understanding both concepts enriches your understanding of grouping problems.

Mastering combinatorics allows you to tackle a wide array of mathematical and practical problems involving different arrangements and selections of objects.

Arrangement Problems

Arrangement problems, also known as permutation problems, involve finding the number of ways to order or arrange a set of objects. In our exercise, arranging the 5 individuals in chairs with Al fixed in the middle highlights a common type of arrangement problem.
Here are some essential aspects:

• Fixed Positions: Sometimes, like with Al, certain positions are predetermined, which simplifies the problem by reducing the number of positions that need arranging.
• Permutations: These are calculated using factorial to determine the arrangements for remaining objects, offering a practical way to solve problems efficiently.

When tackling any arrangement problem, a clear understanding of the constraints, such as fixed positions, aids in simplifying and solving the solution swiftly and accurately.