College Algebra
Tutorial 56: Permutation
Learning Objectives
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
!
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 don’t have this key you will have to enter the definition in as follows:
Either way 7! = 5040.
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
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.
This means there are 20 different 2 letter arrangements.
Practice Problems
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.
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