Types of Permutations Factorials Applications of Permutations Conclusion

MATH 105: Finite Mathematics 6-4: Permutations

Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006 Types of Permutations Factorials Applications of Permutations Conclusion

Outline

1 Types of Permutations

2 Factorials

3 Applications of Permutations

4 Conclusion Types of Permutations Factorials Applications of Permutations Conclusion

Outline

1 Types of Permutations

2 Factorials

3 Applications of Permutations

4 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 Applications of Permutations Conclusion

Outline

1 Types of Permutations

2 Factorials

3 Applications of Permutations

4 Conclusion Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 2! = 2 · 1 = 2 3! = 3 · 2 · 1 = 3 · 2! = 6 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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 Types of Permutations Factorials Applications of Permutations Conclusion

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)! 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! 10 · 9 · 8 · 7 · 6 · 5 · 4! P(10, 6) = = (10 − 6)! 4! = 10 · 9 · 8 · 7 · 6 · 5 = 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! 10 · 9 · 8 · 7 · 6 · 5 · 4! P(10, 6) = = (10 − 6)! 4! = 10 · 9 · 8 · 7 · 6 · 5 = 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

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) 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) Types of Permutations Factorials Applications of Permutations Conclusion

More Permutation Computations

Example Use factorials to compute each permutation.

1 Find P(7, 3)

7! 7 · 6 · 5 · 4! P(7, 3) = = = 7 · 6 · 5 = 210 (7 − 3)! 4! 2 Find P(9, 1)

3 Find P(4, 4) Types of Permutations Factorials Applications of Permutations Conclusion

More Permutation Computations

Example Use factorials to compute each permutation.

1 Find P(7, 3)

7! 7 · 6 · 5 · 4! P(7, 3) = = = 7 · 6 · 5 = 210 (7 − 3)! 4! 2 Find P(9, 1)

3 Find P(4, 4) Types of Permutations Factorials Applications of Permutations Conclusion

More Permutation Computations

Example Use factorials to compute each permutation.

1 Find P(7, 3)

7! 7 · 6 · 5 · 4! P(7, 3) = = = 7 · 6 · 5 = 210 (7 − 3)! 4! 2 Find P(9, 1)

9! 9 · 8! P(9, 1) = = = 9 (9 − 1)! 8! 3 Find P(4, 4) Types of Permutations Factorials Applications of Permutations Conclusion

More Permutation Computations

Example Use factorials to compute each permutation.

1 Find P(7, 3)

7! 7 · 6 · 5 · 4! P(7, 3) = = = 7 · 6 · 5 = 210 (7 − 3)! 4! 2 Find P(9, 1)

9! 9 · 8! P(9, 1) = = = 9 (9 − 1)! 8! 3 Find P(4, 4) Types of Permutations Factorials Applications of Permutations Conclusion

More Permutation Computations

Example Use factorials to compute each permutation.

1 Find P(7, 3)

7! 7 · 6 · 5 · 4! P(7, 3) = = = 7 · 6 · 5 = 210 (7 − 3)! 4! 2 Find P(9, 1)

9! 9 · 8! P(9, 1) = = = 9 (9 − 1)! 8! 3 Find P(4, 4)

4! 4! 4! P(4, 4) = = = = 4 · 3 · 2 · 1 = 24 (4 − 4)! 0! 1 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 Applications of Permutations Conclusion

Outline

1 Types of Permutations

2 Factorials

3 Applications of Permutations

4 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 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 the word “Monday”?

Be Careful! What about the word “fell”? 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 the word “Monday”?

Be Careful! What about the word “fell”? 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 the word “Monday”?

P(6, 6) = 6! = 720

Be Careful! What about the word “fell”? 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 the word “Monday”?

P(6, 6) = 6! = 720

Be Careful! What about the word “fell”? 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 Applications of Permutations Conclusion

Outline

1 Types of Permutations

2 Factorials

3 Applications of Permutations

4 Conclusion Types of Permutations Factorials Applications of Permutations Conclusion

Important Concepts

Things to Remember from Section 6-4

1 Order matters in a permutation

2 r n! Formulas: n with replacement and (n−r)! without.

3 Arranging letters in words: watch out for repetitions! Types of Permutations Factorials Applications of Permutations Conclusion

Important Concepts

Things to Remember from Section 6-4

1 Order matters in a permutation

2 r n! Formulas: n with replacement and (n−r)! without.

3 Arranging letters in words: watch out for repetitions! Types of Permutations Factorials Applications of Permutations Conclusion

Important Concepts

Things to Remember from Section 6-4

1 Order matters in a permutation

2 r n! Formulas: n with replacement and (n−r)! without.

3 Arranging letters in words: watch out for repetitions! Types of Permutations Factorials Applications of Permutations Conclusion

Important Concepts

Things to Remember from Section 6-4

1 Order matters in a permutation

2 r n! Formulas: n with replacement and (n−r)! without.

3 Arranging letters in words: watch out for repetitions! 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