Problem 53 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

At a college library exhibition of faculty publications, three mathematics books, four social science books, and three biology books will be displayed on a shelf. (Assume that none of the books is alike.) a. In how many ways can the ten books be arranged on the shelf? b. In how many ways can the ten books be arranged on the shelf if books on the same subject matter are placed together?

Step-by-Step Solution

Answer

There are 3,628,800 ways to arrange the ten books on the shelf with no restrictions, and there are 14,976 ways to arrange the books on the shelf with books of the same subject placed together.

Step 1: Find the total number of ways to arrange the books

Since there are 10 different books, we can use the formula for the number of permutations of n objects: Number of permutations = n! (n factorial) In this case, n = 10, so the number of permutations is: 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800 There are 3,628,800 ways to arrange the ten books on the shelf with no restrictions. #b. Arranging the books with books of the same subject together#

Step 1: Group the books by subject

We will treat each group of subject books as a single unit: - Math: 3 books - Social Science: 4 books - Biology: 3 books There are 3 groups (subjects) to arrange.

Step 2: Arrange the groups

There are 3! ways to arrange the 3 groups: 3! = 3 × 2 × 1 = 6 There are 6 ways to arrange the groups.

Step 3: Arrange the books within each group

Now we need to find the number of ways to arrange the books within each group: - Math: 3! ways (since there are 3 books in the group) - Social Science: 4! ways (since there are 4 books in the group) - Biology: 3! ways (since there are 3 books in the group)

Step 4: Calculate the total number of arrangements

To find the total number of arrangements, we multiply the number of ways to arrange the groups by the number of ways to arrange the books within each group: Total arrangements = (Number of ways to arrange groups) × (Number of ways to arrange math books) × (Number of ways to arrange social science books) × (Number of ways to arrange biology books) Total arrangements = 6 × 3! × 4! × 3! = 6 × (3 × 2 × 1) × (4 × 3 × 2 × 1) × (3 × 2 × 1) = 6 × 6 × 24 × 6 = 14,976 There are 14,976 ways to arrange the ten books on the shelf with books of the same subject placed together.

Key Concepts

Factorial Notation

Understanding factorial notation is key when solving various mathematical problems that involve permutations and combinations. Factorial notation, represented by an exclamation mark (!), expresses the product of an integer and all the non-zero integers below it. For example, the factorial of 5, written as 5!, is calculated as follows:
5! = 5 × 4 × 3 × 2 × 1 = 120
This notation is particularly useful when we need to find out how many different ways we can arrange a set of objects. In the context of our bookshelf problem, where we have ten unique books, the total possible arrangements are given by 10!, a significantly large number demonstrating the vast possibilities for arrangement. It's important to know that the factorial of zero, 0!, is defined to be 1, which often acts as a multiplicative identity in combinatorial problems.

Arrangement of Objects

When it comes to arranging a certain number of objects, permutations are the go-to concept in combinatorial mathematics. Permutations refer to the different ways in which a set of objects can be arranged in order where the order of arrangement is important. In our example involving books on a shelf, the position of each book matters.
To illustrate further, consider the arrangement of 3 books A, B, and C. These books can be lined up in 3! (which is 3 factorial) ways: ABC, ACB, BAC, BCA, CAB, and CBA. Each arrangement is unique because the order changes. The same principle applies when we arrange 10 books, leading to the vast number of 10! permutations. When solving such problems, it's crucial to clearly define whether each object is distinct or if groupings are involved, as this greatly affects the count of potential arrangements.

Combinatorial Mathematics

The field of combinatorial mathematics deals with the counting, arrangement, and combination of objects. This area of mathematics becomes prominent in questions of probability, statistics, and even computer science. The concepts of permutations and combinations both originate from this domain. While permutations are about ordering objects, combinations focus on selecting groups of objects where the order does not matter.
For example, if we want to select 2 books out of 4 without caring for their order, we can calculate it using combination formula. In the bookshelf problem, when we consider the subject groups together, we are delving right into combinatorial concepts to calculate the total arrangements. By breaking the problem into smaller sections—first by arranging the subject groups and then by arranging the books within each group—it becomes manageable. This divide-and-conquer technique exemplifies the elegant strategies used in combinatorial mathematics to simplify and solve complex counting problems.