Section 5.3: Permutations

 

---

Learning Objectives

• Calculate the value of factorial expressions
• Find the number of permutations of n objects
• Find the number of permutations of n objects taken r at a time
• Find the number of permutations when some objects are alike

---

---

Previously, we studied the Fundamental Counting Principle, which allows us to find the total number of outcomes of multiple independent events by multiplying the choices at each step. For example, 3 shirt choices and 2 pants choices give us 3 \(\cdot\) 2 = 6 possible outfits. In this example, the order of the choices is not significant.

Now we explore situations where order matters, like arranging students in a line or letters in a word. A permutation is an arrangement of objects where changing the order creates a different result. In this section, we will study three ways of counting permutations, each with a distinct formula or method.Prior to that, we need to understand what is called factorial notation.

Introduction to Factorial Notation

When solving permutation problems, you'll notice a specific multiplication pattern that appears repeatedly. Let's examine this pattern by looking in detail at our previous example of the Fundamental Counting Principle.

Example: How many ways can you arrange the letters A, B, C, D?

Solution:

• 1st position: We can choose any of the 4 letters (A, B, C, or D) → 4 choices
• 2nd position: After placing one letter, we have 3 remaining letters to choose from → 3 choices
• 3rd position: After placing two letters, we have 2 remaining letters to choose from → 2 choices
• 4th position: After placing three letters, only 1 letter remains → 1 choice

By applying the Fundamental Counting Principle and multiplying the number of available choices at each step, we find that the total number of possible arrangements equals 4 \(\cdot\) 3 \(\cdot\) 2 \(\cdot\) 1 = 24 different ways.

You can see that this calculation follows a distinct multiplication pattern where we start with the total number of objects (4) and multiply by each consecutive smaller integer until we reach 1. Because this pattern of multiplying a number by every positive integer smaller than itself down to 1 occurs so frequently in mathematics, it has been given a special name and symbol. This pattern is called "factorial" and is written with an exclamation point (!). Below is the definition of the factorial notation.

Definition: Permutation Type #1 - Factorial Notation

Factorial notation \(n!\) represents the product of all positive integers from \(n\) down to \(1\), expressed by the formula:

\[n!=n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot\cdots\cdot3\cdot2\cdot1 \nonumber \]

• "\(n!\)" is pronounced "\(n\) factorial."
• So, \(5!=5\cdot4\cdot3\cdot2\cdot1=120\), and is read: "five factorial."
• Also, we define \(0!=1\).

Example #5.3.1 🤔

How many ways can 8 different books be arranged on a shelf?

---

✅ Solution:

Note that the order of each arrangement is significant. So, ORDER MATTERS here. Also, since, all books are to be arranged, we call this a "basic" or "full" permutation. Here we use the formula \(n!\) where \(n=8\). Thus, there are \(8!=8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=\text{40,320}\) different book arrangements.

Example #5.3.2 🤔

You have 10 favorite songs on your Spotify playlist. How many ways can you play all 10 songs in order?

---

✅ Solution:

Note that the order of each arrangement is significant. So, ORDER MATTERS here. Also, since, all songs are to be arranged, we call this a "basic" or "full" permutation. Here we use the formula \(n!\) where \(n=10\). Thus, there are \(10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=\text{3,628,800}\) different song orders.

Example #5.3.3 🤔

How many different ways can 6 different colored pencils be arranged in a pencil case?

---

✅ Solution:

Note that the order of each arrangement is significant. So, ORDER MATTERS here. Also, since, all pencils are to be arranged, we call this a "basic" or "full" permutation. Here we use the formula \(n!\) where \(n=6\). Thus, there are \(6!=6\cdot5\cdot4\cdot3\cdot2\cdot1=720\) different colored pencil arrangements.

Example #5.3.4 🤔

How many distinct arrangements of the letters in the word "FLORIDA" are possible?

---

✅ Solution:

Note that the order of each arrangement is significant. So, ORDER MATTERS here and notice that all 7 letters are different. Also, since, all letters are to be arranged, we call this a "basic" or "full" permutation. Here we use the formula \(n!\) where \(n=7\). Thus, there are \(7!=7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=\text{5,040}\) different 7-letter distinct arrangements.

Examples #5.3.1 thru #5.3.4 each represent a "basic" or "full" permutation, where all the objects are distinct and each is being arranged in a specific order. Here is a quick summary with the examples and the formula.

Basic or Full Permutations

• An ordered arrangement of ALL distinct objects.
• Example: How many ways can you arrange 5 books on a shelf?
• Example: How many 5-letter words can be made from the letters in the word TEXAS?
• Uses the formula: \(n!\)

Now, what if we wanted to only arrange a few of the distinct objects? Let's look at the following example:

Example: How many ways can you choose a president and a vice-president from a group of 4 different people? Assume that no candidate can fill both offices.

Solution: So, in this case, we are not using all four individuals, since we are only choosing two from the group of four. To better understand this further, let’s give the following names for the group: Amy, Bill, Cassidy, and Daphne. Without loss of generality, assume we choose the president first. There are four candidates for president, and three remaining for vice president, for a total of 4 \(\cdot\) 3 = 12 different ways the two offices may be filled.

Distinct Arrangements of Amy, Bill, Cassidy, & Daphne

Amy & Bill	Bill & Amy	Cassidy & Amy	Daphne & Amy
Amy & Cassidy	Bill & Cassidy	Cassidy & Bill	Daphne & Bill
Amy & Daphne	Bill & Daphne	Cassidy & Daphne	Daphne & Cassidy

for simplicity,