math 105: finite mathematics 6-4:...
TRANSCRIPT
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Types of Permutations Factorials Applications of Permutations Conclusion
MATH 105: Finite Mathematics6-4: Permutations
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
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Types of Permutations Factorials Applications of Permutations Conclusion
Outline
1 Types of Permutations
2 Factorials
3 Applications of Permutations
4 Conclusion
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Types of Permutations Factorials Applications of Permutations Conclusion
Outline
1 Types of Permutations
2 Factorials
3 Applications of Permutations
4 Conclusion
![Page 4: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/4.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Permutations
In this section we will discuss a special counting technique which isbased on the multiplication principle. This tool is called apermutation.
Permutations
A permutation is an ordered arrangement of r objects chosen fromn available objects.
Note:
Objects may be chosen with, or without, replacement. In eithercase, permutations are really special cases of the multiplicationprinciple.
![Page 5: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/5.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Permutations
In this section we will discuss a special counting technique which isbased on the multiplication principle. This tool is called apermutation.
Permutations
A permutation is an ordered arrangement of r objects chosen fromn available objects.
Note:
Objects may be chosen with, or without, replacement. In eithercase, permutations are really special cases of the multiplicationprinciple.
![Page 6: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/6.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Permutations
In this section we will discuss a special counting technique which isbased on the multiplication principle. This tool is called apermutation.
Permutations
A permutation is an ordered arrangement of r objects chosen fromn available objects.
Note:
Objects may be chosen with, or without, replacement. In eithercase, permutations are really special cases of the multiplicationprinciple.
![Page 7: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/7.jpg)
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 0-9with replacement (i.e. numbers may be reused). How many zipcodes are in the set Z of all possible zip codes?
Formula
The number of ways to arrange r items chosen from n withreplacement is:
nr
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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 0-9with replacement (i.e. numbers may be reused). How many zipcodes 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 withreplacement is:
nr
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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 0-9with replacement (i.e. numbers may be reused). How many zipcodes 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 withreplacement is:
nr
![Page 10: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/10.jpg)
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 Sis the set of all possible play schedules, what is c(S)?
Formula
The number of ways to arrange r items chosen from n withoutreplacement is:
P(n, r) =n!
(n − r)!
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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 Sis 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 withoutreplacement is:
P(n, r) =n!
(n − r)!
![Page 12: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/12.jpg)
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 Sis 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 withoutreplacement is:
P(n, r) =n!
(n − r)!
![Page 13: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/13.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Outline
1 Types of Permutations
2 Factorials
3 Applications of Permutations
4 Conclusion
![Page 14: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/14.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
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Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
![Page 16: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/16.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
![Page 17: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/17.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
1! = 1
![Page 18: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/18.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1
1! = 1
2! = 2 · 1 = 2
![Page 19: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/19.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany 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
![Page 20: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/20.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany counting formulas.
Factorial
n! = n · (n − 1) · (n − 2) · . . . · 2 · 1
Example
0! = 1 4! = 4 · 3! = 24
1! = 1
2! = 2 · 1 = 2
3! = 3 · 2 · 1 = 3 · 2! = 6
![Page 21: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/21.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany 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
![Page 22: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/22.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany 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
![Page 23: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/23.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Definition of a Factorial
The formula n! stands for “n factorial”. This will be useful inmany 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)!
![Page 24: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/24.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedulethe plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if Sis the set of all possible play schedules, what is c(S)?
Note:
This may seem more complicated than necessary, but it issometimes useful to have a formula with which to work.
![Page 25: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/25.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedulethe plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if Sis the set of all possible play schedules, what is c(S)?
Note:
This may seem more complicated than necessary, but it issometimes useful to have a formula with which to work.
![Page 26: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/26.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedulethe plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if Sis the set of all possible play schedules, what is c(S)?
P(10, 6) =10!
(10 − 6)!=
10 · 9 · 8 · 7 · 6 · 5 ·��4!
��4!
= 10 · 9 · 8 · 7 · 6 · 5= 151, 200
Note:
This may seem more complicated than necessary, but it issometimes useful to have a formula with which to work.
![Page 27: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/27.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Computing Permutations using Factorials
We can use factorials to compute the number of ways to schedulethe plays mentioned in a previous example.
Example
If there are ten plays ready to show, and 6 time slots available, if Sis the set of all possible play schedules, what is c(S)?
P(10, 6) =10!
(10 − 6)!=
10 · 9 · 8 · 7 · 6 · 5 ·��4!
��4!
= 10 · 9 · 8 · 7 · 6 · 5= 151, 200
Note:
This may seem more complicated than necessary, but it issometimes useful to have a formula with which to work.
![Page 28: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/28.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
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)
![Page 29: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/29.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
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)
![Page 30: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/30.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
More Permutation Computations
Example
Use factorials to compute each permutation.
1 Find P(7, 3)
P(7, 3) =7!
(7 − 3)!=
7 · 6 · 5 · 4!
4!= 7 · 6 · 5 = 210
2 Find P(9, 1)
3 Find P(4, 4)
![Page 31: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/31.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
More Permutation Computations
Example
Use factorials to compute each permutation.
1 Find P(7, 3)
P(7, 3) =7!
(7 − 3)!=
7 · 6 · 5 · 4!
4!= 7 · 6 · 5 = 210
2 Find P(9, 1)
3 Find P(4, 4)
![Page 32: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/32.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
More Permutation Computations
Example
Use factorials to compute each permutation.
1 Find P(7, 3)
P(7, 3) =7!
(7 − 3)!=
7 · 6 · 5 · 4!
4!= 7 · 6 · 5 = 210
2 Find P(9, 1)
P(9, 1) =9!
(9 − 1)!=
9 · 8!
8!= 9
3 Find P(4, 4)
![Page 33: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/33.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
More Permutation Computations
Example
Use factorials to compute each permutation.
1 Find P(7, 3)
P(7, 3) =7!
(7 − 3)!=
7 · 6 · 5 · 4!
4!= 7 · 6 · 5 = 210
2 Find P(9, 1)
P(9, 1) =9!
(9 − 1)!=
9 · 8!
8!= 9
3 Find P(4, 4)
![Page 34: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/34.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
More Permutation Computations
Example
Use factorials to compute each permutation.
1 Find P(7, 3)
P(7, 3) =7!
(7 − 3)!=
7 · 6 · 5 · 4!
4!= 7 · 6 · 5 = 210
2 Find P(9, 1)
P(9, 1) =9!
(9 − 1)!=
9 · 8!
8!= 9
3 Find P(4, 4)
P(4, 4) =4!
(4 − 4)!=
4!
0!=
4!
1= 4 · 3 · 2 · 1 = 24
![Page 35: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/35.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
General Permutation Rules
The last two examples of the previous slide are examples of generalrules for permutations.
Permutation Rule #1
For any n,P(n, 1) = n
Permutation Rule #2
For any n,P(n, n) = n!
![Page 36: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/36.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
General Permutation Rules
The last two examples of the previous slide are examples of generalrules for permutations.
Permutation Rule #1
For any n,P(n, 1) = n
Permutation Rule #2
For any n,P(n, n) = n!
![Page 37: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/37.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
General Permutation Rules
The last two examples of the previous slide are examples of generalrules for permutations.
Permutation Rule #1
For any n,P(n, 1) = n
Permutation Rule #2
For any n,P(n, n) = n!
![Page 38: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/38.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Outline
1 Types of Permutations
2 Factorials
3 Applications of Permutations
4 Conclusion
![Page 39: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/39.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the lettersA,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 howmany ways can this be done?
![Page 40: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/40.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the lettersA,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 howmany ways can this be done?
![Page 41: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/41.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the lettersA,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 howmany ways can this be done?
![Page 42: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/42.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Codes and Committees
Example
How many different 4-letter codes can be formed using the lettersA,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 howmany ways can this be done?
P(7, 3) = 7 · 6 · 5 = 210
![Page 43: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/43.jpg)
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 apool of 6 male and 4 female actors. How many casts are possible?
![Page 44: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/44.jpg)
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 apool of 6 male and 4 female actors. How many casts are possible?
Male: P(6, 2) = 6 · 5 = 30
![Page 45: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/45.jpg)
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 apool of 6 male and 4 female actors. How many casts are possible?
Male: P(6, 2) = 6 · 5 = 30
Female: P(4, 1) = 4
![Page 46: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/46.jpg)
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 apool 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
![Page 47: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/47.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
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 theword “Monday”?
Be Careful!
What about the word “fell”?
![Page 48: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/48.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
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 theword “Monday”?
Be Careful!
What about the word “fell”?
![Page 49: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/49.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
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 theword “Monday”?
P(6, 6) = 6! = 720
Be Careful!
What about the word “fell”?
![Page 50: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/50.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
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 theword “Monday”?
P(6, 6) = 6! = 720
Be Careful!
What about the word “fell”?
![Page 51: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/51.jpg)
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?
![Page 52: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/52.jpg)
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)
P(2, 2)=
4!
2!= 12
Example
In how many ways can the letters in the words “ninny” and“Mississippi” be arranged?
![Page 53: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/53.jpg)
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)
P(2, 2)=
4!
2!= 12
Example
In how many ways can the letters in the words “ninny” and“Mississippi” be arranged?
![Page 54: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/54.jpg)
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)
P(2, 2)=
4!
2!= 12
Example
In how many ways can the letters in the words “ninny” and“Mississippi” be arranged?
P(5, 5)
P(3, 3)=
5!
3!= 20
![Page 55: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/55.jpg)
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)
P(2, 2)=
4!
2!= 12
Example
In how many ways can the letters in the words “ninny” and“Mississippi” be arranged?
P(11, 11)
P(4, 4) · P(4, 4) · P(2, 2)=
11!
4! · 4! · 2!= 34650
![Page 56: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/56.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Outline
1 Types of Permutations
2 Factorials
3 Applications of Permutations
4 Conclusion
![Page 57: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/57.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Important Concepts
Things to Remember from Section 6-4
1 Order matters in a permutation
2 Formulas: nr with replacement and n!(n−r)! without.
3 Arranging letters in words: watch out for repetitions!
![Page 58: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/58.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Important Concepts
Things to Remember from Section 6-4
1 Order matters in a permutation
2 Formulas: nr with replacement and n!(n−r)! without.
3 Arranging letters in words: watch out for repetitions!
![Page 59: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/59.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Important Concepts
Things to Remember from Section 6-4
1 Order matters in a permutation
2 Formulas: nr with replacement and n!(n−r)! without.
3 Arranging letters in words: watch out for repetitions!
![Page 60: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/60.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Important Concepts
Things to Remember from Section 6-4
1 Order matters in a permutation
2 Formulas: nr with replacement and n!(n−r)! without.
3 Arranging letters in words: watch out for repetitions!
![Page 61: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/61.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Next Time. . .
Permutations are the first of two counting rules which build on themultiplication 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
![Page 62: MATH 105: Finite Mathematics 6-4: Permutationsmath.wallawalla.edu/.../slides/finite_chapter_6-4.pdf · Types of Permutations Factorials Applications of Permutations Conclusion MATH](https://reader035.vdocuments.mx/reader035/viewer/2022070713/5ed1e7e43753cc028b5fffdc/html5/thumbnails/62.jpg)
Types of Permutations Factorials Applications of Permutations Conclusion
Next Time. . .
Permutations are the first of two counting rules which build on themultiplication 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