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.