Problem 15 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$P(9,2)$$

Step-by-Step Solution

Answer

The value of \( P(9, 2) \) is 72.

Step 1: Understand Permutation Notation

A permutation, denoted as \( P(n, r) \), refers to the arrangement of \( r \) objects from a set of \( n \) objects, where order matters. Here, we need to find the number of ways to arrange 2 objects from a set of 9.

Step 2: Apply the Permutation Formula

The formula for permutations is \( P(n, r) = \frac{n!}{(n-r)!} \). Plug in the values \( n = 9 \) and \( r = 2 \) into the formula to get \( P(9, 2) = \frac{9!}{(9-2)!} = \frac{9!}{7!} \).

Step 3: Simplify the Factorials

In the permutation formula, simplify \( \frac{9!}{7!} \). Recall that 9 factorial (\( 9! \)) equals \( 9 \times 8 \times 7! \). Therefore, \( \frac{9!}{7!} = \frac{9 \times 8 \times 7!}{7!} = 9 \times 8 \).

Step 4: Compute the Result

Multiply 9 by 8 to find the numerical result of the permutation: \( 9 \times 8 = 72 \).

Key Concepts

Factorial

The factorial of a number, denoted by the exclamation mark (!), is the product of all positive integers up to that number. For example, the factorial of 4 is calculated as follows:

• 4! = 4 × 3 × 2 × 1 = 24

Understanding factorials is crucial because they are used in various mathematical contexts, such as permutations and combinations.
In the context of our exercise, we encountered 9 factorial, which would be expressed as:

• 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Factorials rapidly increase in size, so it is often impractical to compute them fully for large numbers—hence the use of simplification shortcuts. This concept helps us calculate permutations efficiently, as seen when simplifying expressions like

• \( \frac{9!}{7!} = 9 \times 8 \)

Permutation Formula

Permutations are arrangements where the order of elements does matter. The permutation formula, denoted as \( P(n, r) \), helps us calculate the number of possible arrangements of \( r \) objects from a set of \( n \) objects. The permutation formula is expressed as:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

To solve our exercise, we calculated \( P(9, 2) \), where we arranged 2 objects from a set of 9:

• \( \frac{9!}{(9-2)!} = \frac{9!}{7!} = 9 \times 8 \)

This simple formula is foundational in combinatorics and is especially useful in problems where the sequence in which items appear makes a difference.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arrangement, and combination of elements within a set. It deals with choosing elements from a set and arranging them based on given constraints.

• Permutation: The arrangement where order matters, like arranging books on a shelf.
• Combination: The selection where order does not matter, like choosing a committee.

In our problem, we deal with permutations because the order in which we arrange the 2 objects from the set of 9 matters.
The study of combinatorics helps solve real-world problems, like scheduling, network design, or distributing resources efficiently.
Its techniques and formulas simplify complex problems, ensuring we can calculate possible arrangements or selections effectively and efficiently. This demonstrates why understanding permutations is essential in combinatorial calculations.