Back To Course

Math 102: College Mathematics14 chapters | 108 lessons

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Chad Sorrells*

Chad has taught Math for the last 9 years in Middle School. He has a M.S. in Instructional Technology and Elementary Education.

A permutation is a method used to calculate the total outcomes of a situation where order is important. In this lesson, John will use permutations to help him organize the cards in his poker hand and order a pizza.

John is an avid card player. His favorite card game to play is poker. The best part about playing poker for him is the moment when the cards are dealt. John has always been curious about how many different ways he could organize his cards. John seemed to remember that this was called a **permutation**. He researched and found that a permutation is an arrangement of items or events in which order is important. The next time he played poker, he wanted to calculate how many different ways he could organize his cards that he was dealt. The dealer dealt him an Ace of spades, 7 of clubs, 7 of diamonds, Jack of hearts and a 2 of clubs.

To calculate this permutation, John will need to use five blanks to represent the five cards that each player was dealt. John knew that he had five different cards to organize. So, for the first card, he could use any of the five cards. So, John had five choices for the first card. For each subsequent blank, John will have one less choice because he is using cards in the previous blanks. So, in the third blank he will have three choices; in the fourth blank, two choices; and in the fifth blank, one choice.

John now realizes that he has five choices for the first card, four options for the second card, three options for the third card, two options for the fourth card and one option for the fifth card. To find the number of ways that he can organize his cards, he now needs to multiply these numbers together. John multiplied 5 * 4 * 3 * 2 * 1, and the product was 120. John now knows that there are 120 ways to organize his poker hand of cards.

Another way to think about permutations is to understand **factorials**. A factorial is the product of all of the positive integers equal to and less than your number. A factorial is written as a number followed by an exclamation point. For example, to write the factorial of 5, you would write 5!. To calculate the factorial of 5, you would multiply all of the positive integers equal to and less than 5. 5! = 5 * 4 * 3 * 2 * 1. By multiplying these numbers together, we can find that 5! = 120.

Let's look at another example - how would we write and solve the factorial of 10? The factorial of 10 would be written as 10!. To calculate 10!, it would equal 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800.

When writing permutations, we use the notation *n*P*r*, where *n* represents the number of items to choose from, P stands for permutation and *r* stands for how many items you are choosing. To calculate the permutation using this formula, you would use *n*P*r* = *n*! / (*n* - *r*)!.

John is still playing poker and enjoying a lot of success. The dealer asks the group of six players if four of them would like to join a private game. John is excited but curious about how many different ways the four players can be selected. Using the formula *n*P*r*, *n* would represent the total number of players, which is 6. The *r* term would represent the number of players that are being chosen, which is 4. So, this equation would look like 6P4.

To solve this equation, John will calculate 6! and divide it by (6 - 4)!. The first thing John must do is subtract the (6 - 4) in the denominator to get 2. John's equation now looks like 6! / 2!. To calculate 6!, John will multiply 6 * 5 * 4 * 3 * 2 * 1 = 720, and 2! = 2 * 1 = 2. So, we now have 720 / 2, which equals 360. There are 360 ways that four players from John's table can be selected to play at a private table.

Before heading to the private poker game, John stops by the snack bar. The snack bar makes pizzas and has a total of ten toppings. The snack bar sells pizzas with a maximum number of four toppings. How many different types of pizzas can John make?

To calculate the number of choices, you would need to use the equation 10P4 = 10! / (10 - 4)! The first step that John must do is subtract (10 - 4) = 6. So, 10! / (10 - 4)! = 10! / 6!. To calculate 10!, you would multiply 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1. To calculate the 6!, you would multiply 6 * 5 * 4 * 3 * 2 * 1. The easiest way to calculate the total number of choices is to cancel out common terms. Since 6 * 5 * 4 * 3 * 2 * 1 appears on both the top and bottom, these terms can be cancelled out. So, this would leave (10 * 9 * 8 * 7 ) / 1, which equals 5,040 / 1, which equals 5,040. John can choose from 5,040 different types of pizza.

So, let's review what we've learned about permutations. A **permutation** is an arrangement of items or events in which order is important. Permutations help us find the total number of ways that items can be chosen when order does matter. A key tool in calculating permutations is factorials. **Factorials** are written with an exclamation point - for example, *n*!. To find the factorial of a number, multiply all of the positive integers equal to or less than that number. For example, 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040.

To calculate permutations, we use the equation *n*P*r*, where *n* is the total number of choices and *r* is the amount of items being selected. To solve this equation, use the equation *n*P*r* = *n*! / (*n* - *r*)!.

Following this lesson, you'll be able to:

- Express the characteristics of a permutation and a factorial
- Write the equation for calculating permutations
- Solve permutation problems

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 79 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
12 in chapter 13 of the course:

Back To Course

Math 102: College Mathematics14 chapters | 108 lessons

- Go to Logic

- Go to Sets

- Understanding Bar Graphs and Pie Charts 9:36
- How to Calculate Percent Increase with Relative & Cumulative Frequency Tables 5:47
- How to Calculate Mean, Median, Mode & Range 8:30
- Calculating the Standard Deviation 13:05
- Probability of Simple, Compound and Complementary Events 6:55
- Probability of Independent and Dependent Events 12:06
- Either/Or Probability: Overlapping and Non-Overlapping Events 7:05
- Probability of Independent Events: The 'At Least One' Rule 5:27
- How to Calculate Simple Conditional Probabilities 5:10
- Math Combinations: Formula and Example Problems 7:14
- How to Calculate the Probability of Combinations 11:00
- How to Calculate a Permutation 6:58
- Go to Probability and Statistics

- Go to Geometry

- PECT PreK-4: Practice & Study Guide
- ISEE Middle Level: Practice & Study Guide
- ISEE Lower Level: Practice & Study Guide
- ISEE Upper Level: Practice & Study Guide
- California Red Cross Nurse Assistant Competency Evaluation (CNA Test) Training
- Database Management & Data Analytics
- Impact of Lipids on Nutrition
- The Role of Vitamins in Nutrition
- How the Body Handles Nutrients
- Role of Physical Fitness in Nutrition
- What Are WIDA Standards?
- WIDA Can Do Descriptors for Kindergarten
- Demographics for English Language Learners
- Is the TAP Test Hard?
- What is Professional Development for Teachers?
- MTEL Content Test Requirements
- How to Earn Kanban Certification

- Multiple Costing: Definition & Calculation
- Brand Strength & Pricing in Hospitality & Tourism
- Classroom Activities for Introverts
- The Black Legend: Definition & History
- Bermuda Triangle Lesson Plan
- ESL Making Arrangements Vocabulary
- How to Change Your Name: Laws & Process
- Working With Linear Formulas
- Quiz & Worksheet - How Teachers & Administrators Collaborate
- Quiz & Worksheet - Using a Child's Interests to Promote Learning
- Quiz & Worksheet - Analyzing Email Campaigns
- Quiz & Worksheet - Behavioral Theories in Business Communication
- Quiz & Worksheet - Reggio Emilia Educational Approach
- Graphing & Evaluating Equations & Functions Flashcards
- Exponential & Logarithmic Function Flashcards

- Intro to Calculus
- GACE Marketing Education: Practice & Study Guide
- Business Law in Sales
- Group Counseling Strategies: Help & Review
- Business 306: Strategic Human Resources Management
- Factoring with Variables
- Nucleic Acids - DNA and RNA: Help and Review
- Quiz & Worksheet - Using Personality Career Assessments to Choose a Career
- Quiz & Worksheet - International Trade Policy & Strategic Trade Policies
- Quiz & Worksheet - Health Effects Associated with Mineral Consumption
- Quiz & Worksheet - Organizing Your Speech
- Quiz & Worksheet - Organization & Stakeholder Relationships

- Social Information Processing: Definition and Importance in Job Design
- Slavic Countries
- Best Way to Learn Spanish
- What is the AFQT Percentile Score?
- Dinosaur Activities for Kids
- Florida Alternative Teacher Certification
- What is the International Baccalaureate Primary Years Program?
- International Baccalaureate vs. Advanced Placement Tests
- Adult Learning Resources
- Engineering Internships for High School Students
- What is a College Deferral Letter?
- What is the ELM Test?

Browse by subject