Problem 10 - Algebra and Trigonometry

Question

In how many ways can five different mathematics books be placed next to each other on a shelf?

Step-by-Step Solution

Answer

120 ways.

Step 1: Understanding the Problem

We have five different mathematics books that need to be arranged on a shelf. We want to determine the number of possible arrangements.

Step 2: Defining the Arrangement

Arranging books on a shelf is a permutation problem because the order of the books matters.

Step 3: Applying the Permutation Formula

The permutation formula for arranging \( n \) distinct items is \( n! \). In this case, we have 5 books, so we calculate \( 5! \).

Step 4: Calculating 5!

Calculate the factorial of 5: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 5: Solving the Calculation

Perform the multiplication: \( 5 \times 4 = 20 \), \( 20 \times 3 = 60 \), \( 60 \times 2 = 120 \), \( 120 \times 1 = 120 \).

Step 6: Conclusion

Therefore, there are 120 ways to arrange five different mathematics books on a shelf.

Key Concepts

Factorial

The concept of a factorial, often notated as the "!" symbol, is a fundamental mathematical operation. To calculate the factorial of a number means to find the product of all positive integers up to that number. For example, the factorial of 5, written as \( 5! \), is calculated as \( 5 \times 4 \times 3 \times 2 \times 1 \). This results in 120. The factorial function grows rapidly with increasing numbers, making it crucial in the context of permutations and combinations. In permutations, the factorial helps us determine the number of different ways that a set of items can be arranged. Because it multiplies sequential integers, it effectively accounts for all the possible positions that each item can take as others are also arranged around them. Factorials appear often in problems related to arranging objects where the sequence matters.

Arrangement of Distinct Items

When arranging a set of distinct items, each item is unique and can occupy a position different from the others. In our problem, we deal with five unique mathematics books. The question asks us to find out how many ways these books can be laid out on a shelf. This requires considering all possible permutations of these five distinct items. The key here is understanding that because each book is different, their arrangement will lead to various outcomes. We apply the factorial function to determine how many ways these items can be placed in a sequence. - When dealing with distinct items: - Each item occupies a unique space - No repetitions occur Using permutations, specifically expressed with a factorial, ensures we capture all possible sequences for arranging items uniquely.

Order Matters

The phrase "order matters" is paramount in permutation problems, distinguishing them from combinations where order doesn't matter. If we change the sequence of books on a shelf, each arrangement is counted uniquely. For example, placing Book A first and Book B second is different from placing Book B first and Book A second. Therefore, in permutations, each different sequence represents a new permutation. This principle means that the number of permutations increases dramatically with each additional item. Compare arranging two items: Book A followed by Book B is one permutation, while Book B followed by Book A is another. With five books, this concept explains why calculations quickly escalate to 120 possible permutations from just a handful. Understanding that order matters helps to frame why each arrangement of items is unique and must be calculated individually in permutation problems.