# 1.3: Permutations

**At Grade**Created by: Bruce DeWItt

### Learning Objectives

- Know the definition of a permutation
- Be able to calculate the number of permutations using the permutations formula and with technology
- Understand the connection between the Fundamental Counting Principle and permutations

The Fundamental Counting Principle provides us with a tool that allows us to calculate the number of outcomes possible in many situations. What if the situation is a bit more complex? For many situations, the order that we complete a task does not matter. Ordering milk, bacon, and scrambled eggs in that order is the same as ordering bacon, scrambled eggs, and milk. In this case the order that we make our choices wouldn't matter, but there are many situations in which the order that we do things does make a difference.

A **permutation** is a specific order or arrangement of a set of objects or items. What if you wish to call someone on the phone? If I make the call, the order that I punch in the numbers matters so this is an example of a permutation. A good question to ask when deciding if your arrangement is a permutation is **"DOES ORDER MATTER?"** If yes, then you are dealing with a permutation. For example, if you ordered an ice cream sundae and they put the cherry in first, then the chocolate sauce, and then the ice cream, you would probably would not be happy with that particular ice cream sundae. You would likely prefer that they put the ice cream in first, then the chocolate sauce, and then put the cherry on top. Clearly each sundae had the same three ingredients, but they were quite different from one another. Each order that we can make the ice cream sundae is called a permutation.

There is a simple formula for figuring out how many permutations exist when 'r' objects are selected from a set of 'n' objects. The left side of the equation can be read "n P r", just as it looks or "n Permutations of size r".

Recall that the exclamation point is a factorial. For example, \begin{align*}5!=5\times4\times3\times2\times1\end{align*}. Also, be sure to find the permutations command on your calculator.

In our ice cream sundae discussion, 'n' would be 3 because there are 3 items to select from and 'r' would also be 3 because we are going to select all three items. Using the permutations formula, this would be \begin{align*}_3P_3=\frac{3!}{(3-3)!}=\frac{3!}{0!}=\frac{6}{1}=6 \end{align*}. In other words, there are 6 different orders that the ice cream sundae could be made. Note that 0! is equal to 1.

#### Example 1

Suppose you are going to order an ice cream cone with two different flavored scoops. You are going to take a picture of your ice cream cone for use in the school newspaper. The ice cream shop has 5 flavors to choose from; chocolate, vanilla, orange, strawberry, and mint. How many different ice cream cone photos are possible?

#### Solution

The first question to ask is "

Does Order Matter?". If it does, then we are dealing with a permutation question. In this case, the order does make a difference. A chocolate on top of vanilla cone looks different than a vanilla on top of chocolate cone. We have five flavors to pick from, so n=5. We are going to select 2 flavors so r=2. \begin{align*}_5P_2=\frac{5!}{(5-2)!}=\frac{5!}{3!}=\frac{120}{6}=20\end{align*} There 20 different permutations of ice cream cones we could order. The notation representing this situation, \begin{align*}_5P_2\end{align*}, can be read as "Five 'P' Two" or "Five permutations of size Two". Be sure to perform this calculation using your calculator as well.

In the example above, you could have also found your answer using the Fundamental Counting Principle. There were 5 choices for the 1st flavor and then only 4 choices for the 2nd flavor. There are \begin{align*}5\times4=20\end{align*} ice cream cones possible.

#### Example 2

Give the value of \begin{align*}_6 P _3\end{align*} by using the formula for permutations. Verify your solution on your calculator.

#### Solution

\begin{align*}_6 P _3=\frac{6!}{(6-3)!}=\frac{6!}{3!}=\frac{720}{6}=120\end{align*}

#### Example 3

Decide whether each of the situations below involves permutations.

a) A five-card poker hand is dealt from a deck of cards.

b) A cashier must give 3 pennies, 2 dimes, a 5 dollar bill, and a 10 dollar bill back as change for a purchase.

c) A student is going to open a padlock that has a three number combination.

d) A child has red, blue, green, yellow, and orange color crayons and will be coloring a rainbow using each color one time.

#### Solution

a) The order you get your five cards for a poker hand does not matter. If one of your cards was the ace of spades, it didn't matter if it was the first card or the last card dealt.

b) The order that the cashier gives you $15.23 in change does not matter as long as the total is $15.23.

c) The order you put in the three numbers for the combination makes a difference. If the correct combination is 12-27-19, the padlock will not open if you enter 19-12-27 even though the same three numbers are used.

d) The order that the child colors the rainbow does make a difference. The color pattern red, blue, green, orange, yellow will look different than green, blue, red, yellow, orange.

### Problem Set 1.3

#### Exercises

1) Use the formula for Permutations, \begin{align*}_n P _r=\frac{n!}{(n-r)!}\end{align*} to find the value for each expression. Confirm each result by using your calculator.

a) \begin{align*}_8 P _3\end{align*}

b) \begin{align*}_4 P _4\end{align*}

c) \begin{align*}_5 P _3\end{align*}

d) \begin{align*}_5 P _0\end{align*}

2) How many 4 letter permutations can be formed from the letters in word *rhombus*?

3) For a board of directors composed of eight people, in how many ways can a president, vice president, and treasurer be selected?

4) How many different ID cards can be made if there are six digits on a card and no digit can be used more than once?

5) In how many ways can seven different brands of laundry soap be displayed on a shelf in a store?

6) A child has four different stickers that can be placed on a model car in a vertical stack. In how many ways can this be done if each sticker is to be used only one time?

7) An inspector must select three tests to perform in a certain order on a manufactured part. He has a choice of seven tests. How many different ways can he perform three tests?

8) In how many different ways can 4 raffle tickets be selected from 50 tickets if each of the 4 ticket holders wins a different prize?

9) A researcher has 5 different antibiotics to test on 5 different rats. Each rat will receive exactly one antibiotic and no rat will receive the same antibiotic as any other rat. In how many different ways can the researcher administer the antibiotics?

10) There are five violinists in an orchestra. Three of them will be selected to play in a trio with a different part for each musician. In how many ways can the trio be selected?

11) There are five violinists in an orchestra. Four of them will be selected to play in a quartet with a different part for each musician. In how many ways can the quartet be selected?

12) There are five violinists in an orchestra. All five of them will be selected to play in a quintet with a different part for each musician. In how many ways can the quintet be selected?

13) There are five violinists in an orchestra. A piece of music is written so that it can be played with either 3, 4, or 5 violinists. Each musician selected to play this piece will play a different part. In how many ways can a group of at least three musicians be selected? Hint: Use your answers from problems 10), 11) and 12).

14) Decide whether each situation below involves permutations. Briefly explain your answers.

a) Sophia picks three color crayons from a box of 12 crayons to make a picture for her cat, Butterscotch.

b) A five-digit code is needed to open up an electronic lock on a car.

c) Twenty race car drivers must each complete three laps at a race track during a time trial, one after another, in order to establish the order in which the cars will start a race the next day.

d) There are seven steps that a student must follow when preparing cookies during their Family and Consumer Sciences course.

#### Review Exercises

15) Use the Fundamental Counting Principle to determine the number of different ways a person could order a meal if they are to pick one entree from four choices, one side order from three choices, and one drink from four choices.

16) A student wishes to check out three books from the library. She will check out one historical fiction book, one biography, and one book on art history. Build a tree diagram to show how many ways can this be done if there are two historical fiction books, three biographies, and two books on art history that she is considering checking out.

17) How many different outcomes are possible for the total on a roll of two dice if one die has 6 sides and one die has 4 sides?

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