Problem 18 - Algebra for College Students

Question

A bookstore manager wants to make a window display that consists of a mathematics book, a history book, and an economics book in that order. He has 13 different mathematics books, 10 different history books, and 5 different economics books from which to choose. How many different displays are possible?

Step-by-Step Solution

Answer

650 different displays are possible.

Step 1: Identify the total number of choices for each book

Determine the number of options available for each category of book. There are 13 different mathematics books, 10 different history books, and 5 different economics books available for the display.

Step 2: Calculate the total number of arrangements

To find the total number of different displays possible, multiply the number of choices for each type of book. This means calculating the product of the number of mathematics books, history books, and economics books available: \[ 13 \times 10 \times 5 \]

Step 3: Perform the multiplication

Compute the product of the numbers: \[ 13 \times 10 = 130 \] and then \[ 130 \times 5 = 650 \]

Key Concepts

Permutations

In combinatorial mathematics, a permutation is an arrangement of all the members of a set into a particular order. For example, if we have three distinct books, arranging them on a shelf in different orders counts as different permutations. When we want to find how many different ways we can arrange a set of items, each unique order counts as a unique permutation. The exercise involves arranging books in a specific order, and each arrangement is a different permutation.

Multiplication Principle

The multiplication principle, or the counting principle, is a fundamental rule in combinatorial mathematics used to determine the number of possible outcomes in a sequence of events. According to this principle, if you have multiple stages in a process and each stage has a certain number of choices, you find the total number of outcomes by multiplying the number of choices at each stage. For example, in the exercise, the bookstore manager can choose from 13 mathematics books, 10 history books, and 5 economics books. By multiplying these numbers, we find the total number of different displays possible. This principle can be applied to many scenarios to simplify counting complex combinations.

Arrangements

Arrangements refer to the different ways in which a set of items can be organized or displayed. In the context of our exercise, arrangements are the different ways the bookstore manager can place the mathematics, history, and economics books in the display window. Each unique sequence of books (e.g., Math-History-Economics) represents a different arrangement. By considering the number of choices for each category and using the multiplication principle to calculate the total number of arrangements, we determine the number of unique displays possible. This concept emphasizes that order matters, as different orders of the same set lead to different outcomes.