How to Calculate a Permutation

How to Calculate a Permutation

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.

How to Calculate a Permutation

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.

Factorials

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.

Permutation Notation

When writing permutations, we use the notation nPr, 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 nPr = 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 nPr, 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.

Example

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.

Lesson Summary

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