Problem 17 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$P(5,1)$$

Step-by-Step Solution

Answer

The number of ways to arrange 1 item out of 5 is 5.

Step 1: Understand the Permutation Formula

The permutation formula is used to find the number of ways to arrange a set of items. The formula for permutations of selecting \(r\) items from \(n\) items is \(P(n, r) = \frac{n!}{(n-r)!}\). Here, \(n = 5\) and \(r = 1\).

Step 2: Substitute Values into the Formula

Substitute \(n = 5\) and \(r = 1\) into the permutation formula: \(P(5, 1) = \frac{5!}{(5-1)!}\).

Step 3: Simplify the Factorials

Calculate the factorials in the expression. First, compute \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Next, compute \((5-1)! = 4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 4: Perform Division

Divide the factorials calculated in the previous step: \(P(5, 1) = \frac{120}{24} = 5\).

Step 5: Interpret the Result

The value \(P(5, 1) = 5\) represents the number of ways to arrange one item out of five.

Key Concepts

Factorials

Factorials are fundamental in many mathematical concepts, especially in permutations and combinations. A factorial is denoted by an exclamation mark (!), and it's the product of all positive integers up to a given number. For instance:

• The factorial of 5, written as 5!, equals 5 × 4 × 3 × 2 × 1, which is 120.
• Similarly, 4! (four factorial) is 4 × 3 × 2 × 1, which equals 24.

Factorials grow rapidly with larger numbers, making them fascinating yet complex to compute manually for higher numbers. They are crucial for calculating permutations, where the arrangement and order matter, as you'll see in combinatorics.

Arrangements

Arrangements in mathematics refer to the different ways in which a set of items can be ordered or organized. Unlike combinations, arrangements (or permutations) consider the order in which items are placed. For example, organizing 5 books on a shelf in different orders is a matter of permutations.
When calculating permutations, understanding the permutation formula is key, which is \(P(n, r) = \frac{n!}{(n-r)!}\). This formula allows us to determine potential arrangements when selecting \(r\) items from \(n\) total items. In the solved exercise, \(P(5, 1)\) tells us how many different ways 1 item can be chosen and arranged out of 5.
Every arrangement is unique because the order is different, emphasizing why we start with factorials to capture all possible ways of organizing the items.

Combinatorics

Combinatorics is a branch of mathematics dealing with the study of countable discrete structures. It involves understanding the different ways objects can be selected, arranged, and combined under various constraints.
This field includes two main categories:

• Permutations, where the order of selection matters, are used in the provided exercise.
• Combinations, where the order does not affect the outcome.

In our example with permutations, \(P(5, 1)\) demonstrates a simple concept in combinatorics: selecting 1 object from 5 while considering the order. Combinatorics helps solve problems by providing methods to count outcomes effectively, which is essential in fields like probability, algorithm design, and more.