Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Classify each problem according to whether it involves a permutation or a combination. In how many ways can the letters of the word GLACIER be arranged?

Step-by-Step Solution

Answer

This problem involves a permutation since we are considering the order of the letters in the word "GLACIER." There are 7 letters in the word, and using the formula for permutations, 7! (7 factorial), the number of ways the letters can be arranged is 5,040.

Step 1: Determine the number of objects (letters)

The word GLACIER has 7 letters: G, L, A, C, I, E, and R.

Step 2: Calculate the number of permutations

We are arranging all 7 letters, so we will use the formula n! to calculate permutations. In this case, n=7, so the number of permutations is: 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 There are 5,040 ways the letters of the word GLACIER can be arranged.

Key Concepts

Factorial

Factorial is a fundamental concept in mathematics, especially when dealing with permutations and arrangements. In simple terms, the factorial of a non-negative integer, denoted by the symbol '!', is the product of all positive integers up to that number. For example, 4 factorial, written as 4!, is the product of all positive integers up to 4, which equals 4 × 3 × 2 × 1 = 24.
The factorial function grows very quickly, meaning it becomes large for even relatively small numbers. It's a crucial tool for calculating permutations, as it represents the total number of ways items can be arranged in sequence.
When solving a problem that involves arranging a set number of items, calculating the factorial helps in determining the total possible sequences. For the word "GLACIER," the calculation of 7! gives us 5,040 different arrangements of the letters, showcasing how factorial simplifies computing large arrangement possibilities.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing configurations of objects. It helps us understand how we can select or arrange items from a set, which is essential in numerous areas such as probability, computer science, and more.
The main distinction within combinatorics is between permutations and combinations. Permutations consider sequences where the order matters, while combinations focus on selections where the order does not matter.
In our exercise with the word "GLACIER," we calculate how the letters can be arranged in different orders, which is a permutation problem. Understanding combinatorics allows us to apply these principles effectively, whether we're solving problems in mathematics or dealing with practical applications like organizing events or optimizing resources.

Arrangements

Arrangements refer to the specific order or sequence of items in a set. In mathematical terms, how you arrange items is crucial when dealing with problems like permutations, as it impacts the final count of possible sequences.
When calculating arrangements, particularly with permutations, the order of items is key. For instance, in the word "GLACIER," each unique order of the 7 letters is considered a different arrangement, contributing to the total count of 5,040 permutations.
Understanding arrangements is not only essential for solving textbook problems but also for real-world scenarios such as scheduling, seating plans, and logistical operations. Being able to calculate and understand different arrangements enhances our ability to organize and optimize in various aspects of life.