Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

In how many different ways can seven keys be arranged on a key ring if the keys can slide completely around the ring?

Step-by-Step Solution

Answer

720 ways.

Step 1: Understand Ring Arrangement

When arranging items in a circular manner, one item is fixed to avoid counting rotations as distinct permutations. This means we will arrange the remaining items in a linear sequence.

Step 2: Formula for Circular Permutations

The formula for the number of ways to arrange n distinct items in a circle is \( (n-1)! \). Here, we have 7 keys, so we apply this formula.

Step 3: Apply the Formula

Substitute 7 for n in \( (n-1)! \). So, the number of ways is \( (7-1)! = 6! \).

Step 4: Calculate the Factorial

Compute \( 6! \), which is \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 5: Compute the Result

Carrying out the multiplication: \( 6 \times 5 = 30 \), \( 30 \times 4 = 120 \), \( 120 \times 3 = 360 \), \( 360 \times 2 = 720 \), \( 720 \times 1 = 720 \).

Key Concepts

Factorial Calculation

Factorial calculation is a fundamental concept in mathematics, especially in permutation and combination problems. It involves the product of all positive integers up to a certain number. The factorial of a number \( n \) is represented by \( n! \). For example, \( 5! \) means multiplying all whole numbers from 1 to 5 together: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorial calculation simplifies the process of determining how many different ways items can be arranged. It's particularly useful for problems involving combinations and permutations. When applying factorials, remember:

• \( 0! = 1\) by definition.
• Factorials grow very fast, making them useful for large-number computations.

Arranging Objects in a Circle

Arranging objects in a circle adds a unique twist compared to linear arrangement. In circular permutations, the position of one object doesn't change as others are rotated around it. This is why one fixed item is used to prevent counting the same arrangement multiple times during rotations. For any \( n \) distinct objects arranged in a circle, the number of unique arrangements is given by \((n - 1)!\). Here’s why:

• The formula considers fixing one object to break the cycle of rotations.
• This makes the other \( n-1 \) objects freely positionable, calculated by their factorial.

Understanding this concept helps when dealing with non-linear arrangement problems, like seating arrangements around a round table or placing keys on a keyring.

Combinatorial Mathematics

Combinatorial mathematics is a field that deals with counting, arranging, and optimizing the organization of objects. From solving simple puzzles to tackling complex mathematical problems, it provides tools to understand how different arrangements can be formed. The principles of combinatorics rely heavily on permutations and combinations:

• Permutations focus on arranging objects where order matters. Circular permutations are a subset where order is cyclical and requires unique handling through factorial adjustment.
• Combinations are used when the order doesn't matter, typically not applicable to circular arrangements but crucial in selection problems.

Combinatorial mathematics is foundational in computer science, probability, and operations research, among other fields. It offers structured ways to solve problems related to counting and arranging, making it a powerful mathematical discipline.