Types of Permutations
Factorials
Applications of Permutations
MATH 105: Finite Mathematics
6-4: Permutations
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Conclusion
Types of Permutations
Factorials
Outline
1
Types of Permutations
2
Factorials
3
Applications of Permutations
4
Conclusion
Applications of Permutations
Conclusion
Types of Permutations
Factorials
Outline
1
Types of Permutations
2
Factorials
3
Applications of Permutations
4
Conclusion
Applications of Permutations
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations
In this section we will discuss a special counting technique which is
based on the multiplication principle. This tool is called a
permutation.
Permutations
A permutation is an ordered arrangement of r objects chosen from
n available objects.
Note:
Objects may be chosen with, or without, replacement. In either
case, permutations are really special cases of the multiplication
principle.
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations
In this section we will discuss a special counting technique which is
based on the multiplication principle. This tool is called a
permutation.
Permutations
A permutation is an ordered arrangement of r objects chosen from
n available objects.
Note:
Objects may be chosen with, or without, replacement. In either
case, permutations are really special cases of the multiplication
principle.
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations
In this section we will discuss a special counting technique which is
based on the multiplication principle. This tool is called a
permutation.
Permutations
A permutation is an ordered arrangement of r objects chosen from
n available objects.
Note:
Objects may be chosen with, or without, replacement. In either
case, permutations are really special cases of the multiplication
principle.
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations with Replacement
Example
U.S. zip codes consist of an ordering of five digits chosen from 0-9
with replacement (i.e. numbers may be reused). How many zip
codes are in the set Z of all possible zip codes?
Formula
The number of ways to arrange r items chosen from n with
replacement is:
nr
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations with Replacement
Example
U.S. zip codes consist of an ordering of five digits chosen from 0-9
with replacement (i.e. numbers may be reused). How many zip
codes are in the set Z of all possible zip codes?
c(Z ) = permutation of 5 digits from 10 with replacement
= 10 · 10 · 10 · 10 · 10
= 100, 000
Formula
The number of ways to arrange r items chosen from n with
replacement is:
nr
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations with Replacement
Example
U.S. zip codes consist of an ordering of five digits chosen from 0-9
with replacement (i.e. numbers may be reused). How many zip
codes are in the set Z of all possible zip codes?
c(Z ) = permutation of 5 digits from 10 with replacement
= 10 · 10 · 10 · 10 · 10
= 100, 000
Formula
The number of ways to arrange r items chosen from n with
replacement is:
nr
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations without Replacement
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
Formula
The number of ways to arrange r items chosen from n without
replacement is:
n!
P(n, r ) =
(n − r )!
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations without Replacement
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
c(S) = permutation of 6 plays from 10 without replacement
= 10 · 9 · 8 · 7 · 6 · 5
= 151, 200
Formula
The number of ways to arrange r items chosen from n without
replacement is:
n!
P(n, r ) =
(n − r )!
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Permutations without Replacement
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
c(S) = permutation of 6 plays from 10 without replacement
= 10 · 9 · 8 · 7 · 6 · 5
= 151, 200
Formula
The number of ways to arrange r items chosen from n without
replacement is:
n!
P(n, r ) =
(n − r )!
Types of Permutations
Factorials
Outline
1
Types of Permutations
2
Factorials
3
Applications of Permutations
4
Conclusion
Applications of Permutations
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
1! = 1
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
1! = 1
2! = 2 · 1 = 2
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
1! = 1
2! = 2 · 1 = 2
3! = 3 · 2 · 1 = 3 · 2! = 6
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
1! = 1
2! = 2 · 1 = 2
3! = 3 · 2 · 1 = 3 · 2! = 6
4! = 4 · 3! = 24
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
4! = 4 · 3! = 24
1! = 1
5! = 5 · 4! = 120
2! = 2 · 1 = 2
3! = 3 · 2 · 1 = 3 · 2! = 6
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
4! = 4 · 3! = 24
1! = 1
5! = 5 · 4! = 120
2! = 2 · 1 = 2
3! = 3 · 2 · 1 = 3 · 2! = 6
...
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful in
many counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
4! = 4 · 3! = 24
1! = 1
5! = 5 · 4! = 120
2! = 2 · 1 = 2
3! = 3 · 2 · 1 = 3 · 2! = 6
...
n! = n(n − 1)!
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedule
the plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
Note:
This may seem more complicated than necessary, but it is
sometimes useful to have a formula with which to work.
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedule
the plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
Note:
This may seem more complicated than necessary, but it is
sometimes useful to have a formula with which to work.
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedule
the plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
10 · 9 · 8 · 7 · 6 · 5 · 4!
10!
=
(10 − 6)!
4!
= 10 · 9 · 8 · 7 · 6 · 5
P(10, 6) =
= 151, 200
Note:
This may seem more complicated than necessary, but it is
sometimes useful to have a formula with which to work.
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedule
the plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if S
is the set of all possible play schedules, what is c(S)?
10 · 9 · 8 · 7 · 6 · 5 · 4!
10!
=
(10 − 6)!
4!
= 10 · 9 · 8 · 7 · 6 · 5
P(10, 6) =
= 151, 200
Note:
This may seem more complicated than necessary, but it is
sometimes useful to have a formula with which to work.
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
2
Find P(9, 1)
3
Find P(4, 4)
Conclusion
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
2
Find P(9, 1)
3
Find P(4, 4)
Conclusion
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
P(7, 3) =
2
Find P(9, 1)
3
Find P(4, 4)
7!
7 · 6 · 5 · 4!
=
= 7 · 6 · 5 = 210
(7 − 3)!
4!
Conclusion
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
P(7, 3) =
2
Find P(9, 1)
3
Find P(4, 4)
7!
7 · 6 · 5 · 4!
=
= 7 · 6 · 5 = 210
(7 − 3)!
4!
Conclusion
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
P(7, 3) =
2
7!
7 · 6 · 5 · 4!
=
= 7 · 6 · 5 = 210
(7 − 3)!
4!
Find P(9, 1)
P(9, 1) =
3
Find P(4, 4)
9 · 8!
9!
=
=9
(9 − 1)!
8!
Conclusion
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
P(7, 3) =
2
7!
7 · 6 · 5 · 4!
=
= 7 · 6 · 5 = 210
(7 − 3)!
4!
Find P(9, 1)
P(9, 1) =
3
Find P(4, 4)
9 · 8!
9!
=
=9
(9 − 1)!
8!
Conclusion
Types of Permutations
Factorials
Applications of Permutations
More Permutation Computations
Example
Use factorials to compute each permutation.
1
Find P(7, 3)
P(7, 3) =
2
7!
7 · 6 · 5 · 4!
=
= 7 · 6 · 5 = 210
(7 − 3)!
4!
Find P(9, 1)
P(9, 1) =
3
9 · 8!
9!
=
=9
(9 − 1)!
8!
Find P(4, 4)
P(4, 4) =
4!
4!
4!
=
=
= 4 · 3 · 2 · 1 = 24
(4 − 4)!
0!
1
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Conclusion
General Permutation Rules
The last two examples of the previous slide are examples of general
rules for permutations.
Permutation Rule #1
For any n,
P(n, 1) = n
Permutation Rule #2
For any n,
P(n, n) = n!
Types of Permutations
Factorials
Applications of Permutations
Conclusion
General Permutation Rules
The last two examples of the previous slide are examples of general
rules for permutations.
Permutation Rule #1
For any n,
P(n, 1) = n
Permutation Rule #2
For any n,
P(n, n) = n!
Types of Permutations
Factorials
Applications of Permutations
Conclusion
General Permutation Rules
The last two examples of the previous slide are examples of general
rules for permutations.
Permutation Rule #1
For any n,
P(n, 1) = n
Permutation Rule #2
For any n,
P(n, n) = n!
Types of Permutations
Factorials
Outline
1
Types of Permutations
2
Factorials
3
Applications of Permutations
4
Conclusion
Applications of Permutations
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the letters
A,B,C,D,E, and F with no repetition?
Example
A committee of 7 people wisht to select a subcommittee of 3,
including a chairman and secretary for the subcommittee. In how
many ways can this be done?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the letters
A,B,C,D,E, and F with no repetition?
P(6, 4) = 6 · 5 · 4 · 3 = 360
Example
A committee of 7 people wisht to select a subcommittee of 3,
including a chairman and secretary for the subcommittee. In how
many ways can this be done?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the letters
A,B,C,D,E, and F with no repetition?
P(6, 4) = 6 · 5 · 4 · 3 = 360
Example
A committee of 7 people wisht to select a subcommittee of 3,
including a chairman and secretary for the subcommittee. In how
many ways can this be done?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the letters
A,B,C,D,E, and F with no repetition?
P(6, 4) = 6 · 5 · 4 · 3 = 360
Example
A committee of 7 people wisht to select a subcommittee of 3,
including a chairman and secretary for the subcommittee. In how
many ways can this be done?
P(7, 3) = 7 · 6 · 5 = 210
Types of Permutations
Factorials
Applications of Permutations
Conclusion
More Committees
Example
A play involving 2 male and 1 female parts is to be cast from a
pool of 6 male and 4 female actors. How many casts are possible?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
More Committees
Example
A play involving 2 male and 1 female parts is to be cast from a
pool of 6 male and 4 female actors. How many casts are possible?
Male:
P(6, 2) = 6 · 5 = 30
Types of Permutations
Factorials
Applications of Permutations
Conclusion
More Committees
Example
A play involving 2 male and 1 female parts is to be cast from a
pool of 6 male and 4 female actors. How many casts are possible?
Male:
P(6, 2) = 6 · 5 = 30
Female:
P(4, 1) = 4
Types of Permutations
Factorials
Applications of Permutations
Conclusion
More Committees
Example
A play involving 2 male and 1 female parts is to be cast from a
pool of 6 male and 4 female actors. How many casts are possible?
Male:
P(6, 2) = 6 · 5 = 30
Female:
P(4, 1) = 4
Combined:
P(6, 2) · P(4, 1) = 30 · 4 = 120
Types of Permutations
Factorials
Applications of Permutations
Arranging Letters in a Word
One application of permutations is rearranging letters in a word.
Example
How many new “words” can be formed from the letters in the
word “Monday”?
Be Careful!
What about the word “fell”?
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Arranging Letters in a Word
One application of permutations is rearranging letters in a word.
Example
How many new “words” can be formed from the letters in the
word “Monday”?
Be Careful!
What about the word “fell”?
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Arranging Letters in a Word
One application of permutations is rearranging letters in a word.
Example
How many new “words” can be formed from the letters in the
word “Monday”?
P(6, 6) = 6! = 720
Be Careful!
What about the word “fell”?
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Arranging Letters in a Word
One application of permutations is rearranging letters in a word.
Example
How many new “words” can be formed from the letters in the
word “Monday”?
P(6, 6) = 6! = 720
Be Careful!
What about the word “fell”?
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Words with Repeated Letters
Example
In how many ways can the letters in the word “fell” be arranged?
Example
In how many ways can the letters in the words “ninny” and
“Mississippi” be arranged?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Words with Repeated Letters
Example
In how many ways can the letters in the word “fell” be arranged?
P(4, 4)
4!
=
= 12
P(2, 2)
2!
Example
In how many ways can the letters in the words “ninny” and
“Mississippi” be arranged?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Words with Repeated Letters
Example
In how many ways can the letters in the word “fell” be arranged?
P(4, 4)
4!
=
= 12
P(2, 2)
2!
Example
In how many ways can the letters in the words “ninny” and
“Mississippi” be arranged?
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Words with Repeated Letters
Example
In how many ways can the letters in the word “fell” be arranged?
P(4, 4)
4!
=
= 12
P(2, 2)
2!
Example
In how many ways can the letters in the words “ninny” and
“Mississippi” be arranged?
P(5, 5)
5!
=
= 20
P(3, 3)
3!
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Words with Repeated Letters
Example
In how many ways can the letters in the word “fell” be arranged?
P(4, 4)
4!
=
= 12
P(2, 2)
2!
Example
In how many ways can the letters in the words “ninny” and
“Mississippi” be arranged?
P(11, 11)
11!
=
= 34650
P(4, 4) · P(4, 4) · P(2, 2)
4! · 4! · 2!
Types of Permutations
Factorials
Outline
1
Types of Permutations
2
Factorials
3
Applications of Permutations
4
Conclusion
Applications of Permutations
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Important Concepts
Things to Remember from Section 6-4
1
Order matters in a permutation
2
Formulas: nr with replacement and
3
Arranging letters in words: watch out for repetitions!
n!
(n−r )!
without.
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Important Concepts
Things to Remember from Section 6-4
1
Order matters in a permutation
2
Formulas: nr with replacement and
3
Arranging letters in words: watch out for repetitions!
n!
(n−r )!
without.
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Important Concepts
Things to Remember from Section 6-4
1
Order matters in a permutation
2
Formulas: nr with replacement and
3
Arranging letters in words: watch out for repetitions!
n!
(n−r )!
without.
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Important Concepts
Things to Remember from Section 6-4
1
Order matters in a permutation
2
Formulas: nr with replacement and
3
Arranging letters in words: watch out for repetitions!
n!
(n−r )!
without.
Conclusion
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Next Time. . .
Permutations are the first of two counting rules which build on the
multiplication principle. In the next section, we will introduce
“combinations” in which we care only about the objects selected,
and not the order in which the selection is made.
For next time
Read Section 6-5 (pp 343-349)
Do Problem Sets 6-4 A,B
Types of Permutations
Factorials
Applications of Permutations
Conclusion
Next Time. . .
Permutations are the first of two counting rules which build on the
multiplication principle. In the next section, we will introduce
“combinations” in which we care only about the objects selected,
and not the order in which the selection is made.
For next time
Read Section 6-5 (pp 343-349)
Do Problem Sets 6-4 A,B