**College Algebra**

**Tutorial 56: P****ermutation**

**Learning Objectives**

After completing this tutorial, you should be able to:

- Use permutations to solve a counting problem involving order.

**Introduction**

In this tutorial we will be going over
permutations. Permutations
are an off shoot of the Fundamental Counting Principle. If you need a
review
on the Fundamental Counting Principle, feel free to go to **Tutorial
55: The Fundamental Counting Principle.** Permutations
specifically
count the number of ways a task can be arranged or ordered. I think you
are ready to go off into the wonderful world of permutations, have fun!

** Tutorial**

**Factorial**

**!**

The factorial symbol is the exclamation point: !

So if I wanted to write 7 factorial it would be written as 7!.

**In general, n! = n(n - 1)(n -
2)(n - 3)...(1)**

**Most, (if not all), of you will have a factorial key
on your calculator.
It looks like this: !**

**If you have a graphing calculator, it will be hidden
under the MATH
menu screen and then select your Probability screen - there you should
find !**

Some calculators don’t have one, so I will show you how to simplify the problems in case you don’t have that key on your calculator.

**0! Has a special
definition attached
with it. 0! = 1**

If you have a ! key on your calculator you simple press
7 and then
! and in some cases you may have to also press your enter or = key.

If you don’t have this key you will have to enter the definition in as follows:

7! = (7)(6)(5)(4)(3)(2)(1) = 5040

**Either way 7! = 5040.**

We needed to know about factorial because it is used
the formula for
permutation, which is our next topic.

An **ORDER **of arrangements
of *r *objects,
without repetition, selected from *n* distinct
objects is called a permutation of *n* objects
taken *r* at a time, and is denoted as

For example, you may want to know how many ways to pick a 1st, 2nd, and 3rd place winner from 10 contestants. Since you are arranging them in order, you could use a permutation to do this. Or if you were wanting to know how many ways your committee could pick a president, a vice president, a secretary, and a treasurer, you could use permutations.

**First we need to find n and r : **

If you said ** n is
8 **you are correct!!!
There are 8 CD's in this problem.

*r* is the number of
CD’s we are using at
a time. What do you think *r* is?

If you said ** r is
8**, pat yourself
on the back!! We are arranging all 8 CD's on the shelf.

**Putting this into the permutation formula we get:**

***0! = 1 **

***Expand out 8!**

If you don't have a factorial key, you can simplify it as shown above and then enter it in. It is probably best to simplify it first, because in some cases the numbers can get rather large, and it would be cumbersome to multiply all those numbers one by one.

**Wow, this means there are 40320 different ways to
arrange those 8
CD’s, that’s a lot.**

**First we need to find n and r :**

If you said ** n is
10 **you are correct!!!
There are 10 teams in this problem.

*r* is the number of
teams we are ranking
at a time. What do you think *r* is?

If you said ** r is
10**, pat yourself
on the back!! We are ranking all 10 teams.

**Putting this into the permutation formula we get:**

***0! = 1 **

***Expand out 10!**

If you don't have a factorial key, you can simplify it as shown above and then enter it in. It is probably best to simplify it first, because in some cases the numbers can get rather large, and it would be cumbersome to multiply all those numbers one by one.

**Wow, this means there are 3,628,800 different ways to
rank those
10 teams, that’s a lot.**

**First we need to find n and r :**

If you said ** n is
20 **you are correct!!!
There are 20 members in this problem.

*r* is the number of
members we are selecting
for offices at a time. What do you think *r* is?

If you said ** r is
3**, pat yourself
on the back!! There are 3 offices.

**Putting this into the permutation formula we get:**

***Expand 20! until it gets to
17! ( which is
the ! in den)**

***Cancel out 17!'s**

If you don't have a factorial key, you can simplify it as shown above and then enter it in. It is probably best to simplify it first, because in some cases the numbers can get rather large, and it would be cumbersome to multiply all those numbers one by one.

**Wow, this means there are 6840 different ways to
select the three
officers, that’s a lot.**

**First we need to find n and r :**

If you said ** n is
5 **you are correct!!!
There are 5 letters in TEXAS.

*r* is the number of
letters we are using
at a time. What do you think *r* is?

If you said ** r is
2**, pat yourself
on the back!! We are using 2 letters at a time.

**Putting this into the permutation formula we get:**

***Expand 5! until it gets to 3!
( which is the
! in den)**

***Cancel out 3!'s**

**This means there are 20 different 2 letter
arrangements.**

** Practice Problems**

These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. **Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1c:Solve using permutations.1a. A company issues a questionnaire whereby each employee must rank the 5 items with which he or she is most satisfied. The items are wages, work environment, vacation time, job security, supervisors, health insurance, break time, and retirement plan.The ranking is to be indicated by the numbers 1, 2, 3, 4 and 5, where 1 indicates the item involving the greatest satisfaction and 5 the least. In how many ways can an employee answer this questionnaire?

(answer/discussion to 1a)1b. A key pad lock has 10 different digits, and a sequence of 5 different digits must be selected for the lock to open. How many key pad combinations are possible?

(answer/discussion to 1b)1c. In how many ways can 7 books be arranged on a shelf?

(answer/discussion to 1c)

Need Extra Help on these Topics?

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on May 20, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.