Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, compute the value of the expression. $$ P(8,4) $$

Step-by-Step Solution

Answer

The value of \( P(8, 4) \) is 1680.

Step 1: Understand the Permutation Notation

The notation \( P(n, r) \) represents the number of permutations of \( n \) items taken \( r \) at a time. In this problem, we need to calculate \( P(8, 4) \).

Step 2: Use the Permutation Formula

The formula for permutations is \[ P(n, r) = \frac{n!}{(n-r)!} \]. In our case, \( n = 8 \) and \( r = 4 \). Thus, we have \[ P(8, 4) = \frac{8!}{(8-4)!} = \frac{8!}{4!} \].

Step 3: Calculate the Factorials

Calculate each factorial separately. - \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \).- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 4: Divide the Factorials

Use the results from the previous step to calculate the value of the permutation:\[ P(8, 4) = \frac{40320}{24} \].Perform the division to find:\[ P(8, 4) = 1680 \].

Key Concepts

Permutation Formula

Permutations are all about arrangements. They refer to the different ways you can arrange a set number of items. In mathematics, the permutation formula is a tool to figure out how many such arrangements are possible.The formula itself is quite straightforward. It is given by:\[ P(n, r) = \frac{n!}{(n-r)!} \]where:

• \(n\) is the total number of items you have
• \(r\) is the number of items to arrange

This formula essentially reduces the problem to calculating two factorials and dividing them. In the example \(P(8, 4)\), it tells us we're looking at all the ways to arrange 4 items out of a total of 8 available.By applying this formula, you can solve a wide range of problems involving order and arrangement, like seating arrangements or password generation. This is why permutations are a fundamental part of combinatorics and algebra.

Factorials

Factorials appear frequently in permutations, and they are a central part of calculating arrangements. A factorial, represented by an exclamation mark (!), is the product of all positive integers up to a given number. For instance, the factorial of 5 (written as \(5!\)) is calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]When you work with permutations, you'll often deal with two types of factorials:

• \(n!\), the factorial of the total number of items
• \((n - r)!\), the factorial of the difference between the number of items and the number you wish to arrange

Factorials grow very fast, making them important in calculations involving large numbers. In the solution for \(P(8, 4)\), you calculate \(8!\) and \(4!\) to find that \(P(8, 4)\) involves dividing the large number 40320 by 24 to get 1680.Understanding factorials is crucial because they simplify complex multiplication problems into concise expressions, which is extremely helpful when dealing with permutations and combinations.

Combinatorics

Combinatorics is the field of mathematics that deals with counting, arrangement, and combination. It's a fascinating subject full of intricate and engaging problems. Within this field, permutations are one of the primary concepts.Combinatorics helps us understand how to count without listing each item individually, which is incredibly useful for large sets. Permutations in combinatorics specifically deal with ordered arrangements of distinct objects and answer questions like "In how many ways can we select and arrange a subset of items from a larger set?"The example of \(P(8, 4)\) is a classic problem in combinatorics: selecting 4 items from 8 and finding all possible arrangements. In broader terms, combinatorics explores topics such as:

• Permutations and combinations
• Graph theory
• Counting techniques

By studying combinatorics, students not only learn to solve simple arrangement problems but also understand complex structures and patterns in data and mathematical systems. It’s a skillset that lays the foundation for many advanced areas in mathematics and applied sciences.