Problem 31 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. \(P(5,3)\)

Step-by-Step Solution

Answer

The value of \( P(5,3) \) is 60.

Step 1: Understand the permutation formula

The formula to find permutations is given by \( P(n, r) = \frac{n!}{(n-r)!} \). In this case, \( n = 5 \) and \( r = 3 \). We need to determine how many ways we can arrange 3 items out of 5.

Step 2: Calculate the factorial for the numerator

Calculate \( n! \), which is \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 3: Calculate the factorial for the denominator

Calculate \( (n-r)! \), which is \( (5-3)! = 2! = 2 \times 1 = 2 \).

Step 4: Complete the permutation calculation

Plug the values into the permutation formula: \( P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60 \). So, there are 60 ways to arrange 3 items out of 5.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. A factorial, denoted by the exclamation mark (e.g., \( n! \)), represents the product of all positive integers up to a certain number \( n \). In simpler terms, to find \( n! \), you multiply all whole numbers from \( 1 \) to \( n \). For example:

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

This concept is important because it helps in calculating the number of ways in which items can be arranged or ordered.
Factorials grow quite rapidly. Even with a small increase in \( n \), the result becomes significantly larger. For instance:

• \( 3! = 6 \)
• \( 4! = 24 \)
• \( 5! = 120 \)

When doing calculations involving permutations, factorials are often used in both the numerator and denominator to simplify expressions and find the total number of permutations.

Permutation Formula

The permutation formula is a powerful tool used to determine how many different ways items can be arranged. It is usually expressed as \( P(n, r) \), which represents the number of permutations of \( n \) items taken \( r \) at a time. The formula is given by:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

Here, \( n! \) calculates all possible arrangements of \( n \) items, while \( (n-r)! \) accounts for the fact we are only choosing \( r \) items instead of using all \( n \).

For instance, consider the problem of computing \( P(5,3) \), which asks how many ways you can arrange 3 items selected from a total of 5 items. You substitute the values into the permutation formula:

• Calculate \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)
• Calculate \( (5-3)! = 2! = 2 \times 1 \)

Using the formula:

\[ P(5,3) = \frac{5!}{2!} = \frac{120}{2} = 60 \]

This means there are 60 different ways to arrange 3 items from a group of 5.

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of elements within a set. It provides the fundamental tools needed for tackling problems involving permutations and combinations.

In particular, combinatorics helps us to understand different scenarios where order matters. For instance, permutations focus on the arrangement of items where the sequence is important. Combinatorics encompasses tasks such as:

• Determining the number of ways to arrange a subset of items.
• Calculating combinations where the order does not matter (although not used in this exercise).
• Analyzing complex problems in probability, graph theory, and more.

In the exercise \( P(5,3) \), combinatorics allows us to determine how many distinct sequences we can create from selecting and ordering 3 items out of a total of 5. This is an example of how combinatorics is not just about numbers, but also about understanding patterns and possibilities in arranging objects.