# What is Permutations: Definition and 292 Discussions

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.
Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.
The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.
Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3,1,2) mentioned above is described by the function

α

{\displaystyle \alpha }
defined as:

α
(
1
)
=
3
,

α
(
2
)
=
1
,

α
(
3
)
=
2

.The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition (performing two given rearrangements in succession), which results in another rearrangement. As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set

{
1
,
2
,

,
n
}

{\displaystyle \{1,2,\ldots ,n\}}
that are considered for studying permutations.
In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

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1. ### Bishops and permutations on chessboard

Let ##n## be a positive integer. Initially, a bishop is placed in each square of the top row of a ##2^n \times 2^n## chessboard; those bishops are numbered from ##1## to ##2^n## from left to right. A jump is a simultaneous move made by all bishops such that each bishop moves diagonally, in a...
2. ### I About permutation acting on the Identity matrix

Question: Let ##\sigma\in S_n## be a permutation and ##T_{\sigma}## be the matrix we obtain from ##I## by appling ##\sigma## on the raws of ##I## (I.e ##\sigma## acts on the rows of ##I##) . Then: 1. ##\det(T_{\sigma}) = sgn(\sigma) ## and 2. ##T_{\sigma} T_{\tau} =T_{\sigma\circ \tau}##, for...
3. ### Mastering Combinatorics: Exploring the Formula for Permutations

My combinatorics professor has a MA, PhD from Princeton University. On our test, she asked I handwrote, but transcribed in Latex, my answer below. How can I improve this? What else should I've written? Professor awarded me merely 50%. She wrote
4. ### I Cycles from patterns in a permutation matrix

In a permutation matrix (the identity matrix with rows possibly rearranged), it is easy to spot those rows which will indicate a fixed point -- the one on the diagonal -- and to spot the pairs of rows that will indicate a transposition: a pair of ones on a backward diagonal, i.e., where the...
5. ### Solve this problem that involves permutations

Tricky questions ; Ok in my approach; [..., .... , ...2...] This can be filled in ##3×3×1=9## ways [..., .... , ...4...] This can be filled in ##3×3×1=9## ways [.... , ...4...] This can be filled in ##4×1=4## ways [.... ...
6. ### MHB Solving Permutations Question with Restrictions

Hello all Please look at this questions: What is the number of permutations for creating a code of 3 digits from the digits 1,2,3,...,9 , such that every digit is equal or larger from the previous one ? I know that if I wanted the number of permutations without restrictions it would be...
7. ### B Optimizing permutations of hero traits in a computer game

I have a game where heroes have a set of traits, or abilities. The level of the abilities are raised in two ways, by banner cards and/or by leveling the hero. The Banner cards and heroes don't match perfectly, rather a banner card can match 1 or 2 (sometimes 3) abilities of the heroes abilities...
8. ### Prob/Stats Books on Combinatorics, Permutations and Probability

Hello! I am looking for textbooks to relearn Combinatorics, Permutations Combinations and Probability and also Matrix algebra( decomposition, etc). I had done these many years ago and the course/books provided to me at that time weren't that great. So I want to relearn this with a more...
9. ### I Permutations written as product of 2-cycles

I'm trying to learn Group Theory from Gallian's book. When I reached the chapter for permutation groups, the author gives an example that we can write (12345) as (15)(14)(13)(12). I immediately recognized that this should always work (I proved it later.) Then author says we can write : (12345)...
10. ### Algebra High school courses on Permutations and combinations

Can you give me some high school papers or courses on p and c . I have a good source for problems but need a concise and compact course covering concepts. Thanks!
11. ### MHB #s of Combinations and Permutations of lines?

Hello All, See picture below: There exist an infinite plane with infinite number of dots. For sake of argument, let's assume they are 1 inch away from each other. However, below(on your far left) you can see 3 lines already made. The last line is the yellow one. What you see on the left, are...
12. ### List all pairs of permutations with repetition with the given conditions

Let us consider two sequences: $$n_k \in \Omega,\,k=1,2,...K,$$ $$m_k \in \Omega,\,k=1,2,...K,$$ where $$\Omega:=\{n\in\mathbb{N}|\,n\leq K\}.$$ Let us define $$\sigma_n:=\sum_{k=1}^K k\, n_k,\,\sigma_m:=\sum_{k=1}^K k\,m_k$$ The task is to find all possible ##(n_k,\,m_k)## pairs such that...
13. ### Stacking rings on fingers by decreasing stack size (permutations)

When she is stacking ##5## rings, then there are ##5P5## configurations when it comes to arranging rings, and each configuration can be arranged on her fingers in ##5P1## ways (choose one finger from 5 to put that configuration on). When she is stacking ##4## rings, then we have two objects; a...

34. ### High Temperature Limit of Entropy in a Two Level System

Homework Statement Sounds like a physics problem but I'm sure of the physics, stuck on the maths. At high T a two level system has ##\frac{N}{2}## particles in each level. If entropy is given by ##S = k\ln(\Omega)##, where ##\Omega## is the number of ways of getting ##\frac{N}{2}## particles...
35. ### Permutations (with repetitions) problem

Homework Statement [/B] The question is phrased in the following way: There are 6 jobs to be assigned to 5 people. Each job is assigned only to one person, and each person must have at least one job. How many different arrangements are there? Homework Equations In general, I would approach a...
36. ### B How are permutations and probability related?

This may already be widely taught and I could be stating the obvious here, but I noticed how closely related permutations and probability are, and this gives an intuitive way to think about permutations. For example, take a deck of 52 cards. How many possible permutations are there for the...
37. ### B Need help in permutations and combinations

Hello, I face problems sometimes in identifying the maths of permutation and combination.Can anyone please tell me an easy way to identify quickly whether the math is about permutation or combination? Thank you, Shafia.
38. ### MHB Probability of Combinations and Permutations

Hi, I'm a mom trying to help my son understand why he got answers wrong on his online math program. He is taking Geometry, but the last unit in the class is an introduction to Probability and Statistics. After re-reviewing the lesson and re-working the problems he got wrong, we were able to...
39. ### I How Are Permutations Calculated in a Circular Seating Arrangement?

2 boys and 3 girls are to be seated round a table with 5 seats. Each child occupies exactly one seat. In how many ways can this be done if (a) the 2 boys must be seated together (b) same as (a) but this time the seats are numbered Solution (a) ##\frac{4!}{4}2!## (b) ##\frac{4!}{4}2!\times 5##...
40. ### Counting permutations of a string with repeating characters

The problem statement: How many five-letter strings of capital letters have a letter repeated twice in a row? For example, include ABCCA and AAABC and ABBCC but not ABCAD. The attempt at a solution: First, let's break down how we would perform the selection of a string that meets the...
41. ### Proof involving group of permutations of {1,2,3,4}.

Homework Statement Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##: ##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...
42. ### Permutations of (abc)(efg)(h) in S7

Homework Statement How many distinct permutations are there of the form (abc)(efg)(h) in S7? Homework Equations 3. The Attempt at a Solution [/B] since we have 7 elements I think for the first part it should be 7 choose 3 then 4 choose 3. And then we multiply those together.
43. ### How to Factorize a Large Number for Permutations?

Hey guys , Could anyone here tell me the easiest way to solve for n , nP7 = 604800 , the traditional way (I'm currently using) is to divide 604800 by 10 and then 9 and so on until I get 1 as a result of that division , The problem is this way isn't helpful with all permutations I have in my...
44. ### How do I calculate permutations of a multi-set with limited elements?

[mentor note: THis is not a homework assignment. It is for a work project.] I need a formula that is probably based on permutations of multi-set. Except in my case you will not use up all elements of the sets, only some of them. For example I have the following sets: {1,1,1}{2}{3}...
45. ### The equivalence of a set and its permutations.

The following is from an introduction to groups. It is not clear to me why the authors bothered to introduce the subset ##\mathfrak{Q}\subseteq \mathfrak{R}## and a subset ##\mathfrak{K}\subseteq \mathfrak{S}^{\mathfrak{R}}## into the discussion. (3) seems to follow trivially from the...
46. ### Application in permutations group

Hello I am studying for my exam and there's a question that i don't know how to solve, I have some difficulties with symmetric/permutations groups 1. Homework Statement Consider a finite group of order > 2. We write Aut(G) for the group of automorphisms of G and Sg for the permutations group...
47. ### Program to list specific permutations of 5 digits

Not long ago, a member posted a problem in the Homework section, concerned with determining which of the five-digit permutations of 4, 5, 6, 8, and 9 were divisible by 8. I first thought about writing some C code to figure this out, but when I discovered that Python has some functions that...
48. ### Prob/Stats Good permutations and combinations problems book

I want to get better at this topic and I have trouble finding good questions apart from past year exam questions. I am currently an A- Level student, so can someone recommend me a good book full of challenging questions at my level? To give you some idea of the types of questions in the exam...
49. ### Cyclic Permutations: εijk, Even or Odd?

εijk is the permutation symbol and cyclic permutations, for example 123→231→312, are always even, thus ε123=ε231=ε312=+1, but: ε132=ε213=ε321=-1 I understand the first 2, but ε321 is even, no? and also all this series is cyclic, it's not all even and...
50. ### Permutations and combinations

Homework Statement Out of 7 women and 4 men you need to choose delegation. On how many ways you can choose delegation that consist of: a) five people - 3 women and 2 men b)any number of people, but with equal number of men and women c) five people with at least 3 women d) five peope where one...