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


{\displaystyle \alpha }
defined as:




{\displaystyle \alpha (1)=3,\quad \alpha (2)=1,\quad \alpha (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



{\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. HighPhy

    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. A

    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. 1

    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. nomadreid

    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. chwala

    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. L

    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. L

    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. C

    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. Kaguro

    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. sahilmm15

    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. L

    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. Robin04

    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. archaic

    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...
  14. M

    MHB Determine the cycle decomposition of the permutations

    Hey! :o We have the following permutations in $\text{Sym}(14)$ : - $\pi_1=(1 \ 2\ 4 \ 9)\circ(1 \ 3)\circ (6 \ 8\ 12)$ - $\pi_2=(2 \ 4\ 5 \ 8\ 7)\circ (1 \ 12 \ 6)\circ \ (13 \ 14)$ - $\pi_3=(1 \ 4 \ 5\ 8 \ 11)\circ (2 \ 4\ 6 \ 5 \ 1)$ 1. Determine the cycle decomposition of $\pi_1...
  15. Monoxdifly

    MHB The number of possible permutations for the assessment

    There are 4 questions in the assessment. In order not to repeat questions in an assessment, question 1 could be any of 13 questions, question 2 any of 21 questions, question 3 anyone of 14 questions and question 4 anyone of 12 questions. How do I calculate the number of possible permutations for...
  16. I

    Optimized Algorithm to Find All 720 Random Permutations of [0, 6)

    Summary: Algorithm do find all random permutations of n=[0,6) Hi, The following algorithm gives 6 out of the 720 random permutations of integers in the range [0, 6). t=0..5 // permutation i=0..5 // sequence index n =...
  17. C

    How many ways can 5 prizes be distributed among 4 boys and 3 dividers?

    the first boy has 6 ways, either he wins 0 or 1 or 2 or 3 or 4 or 5 ways According to the first boy, the second boy also has 0 or 1 or 2 or 3 or 4 or 5 chances According to the first and second boy, the other boy also has 0 or 1 or 2 or 3 or 4 or 5 chances. So, the answer should be 4(either of...
  18. Kaushik

    B Permutations and combinations

    Summary: Is 'Permutation and combination' hard to understand ? I am facing difficulties in understanding this concept. So I would like to ask some questions to enhance my knowledge in this topic. My questions are- Does it require time to grasp this concept? Is it normal to get confused? How...
  19. Math Amateur

    I Permutations and Determinants .... Walschap, Theorem 1.3.1 ...

    I am reading Gerard Walschap's book: "Multivariable Calculus and Differential Geometry" and am focused on Chapter 1: Euclidean Space ... ... I need help with an aspect of the proof of Theorem 1.3.1 ... The start of Theorem 1.3.1 and its proof read as follows: I tried to understand how/why...
  20. A

    Other Highschool Book for Permutations & Combinations

    Can someone please tell me what good book for a high school student on permutation and combination? Thank you
  21. ubergewehr273

    Finding no. of combinations for the situation

    Homework Statement Refer the image Homework Equations Equations for permutations and combinations The Attempt at a Solution Let x be the no. of questions that turned out to be correct. So total score will be 3x-(10-x)=4x-10. The value of this expression must be from the given set and since x...
  22. resurgance2001

    Statistics Permutations and Combinations

    Homework Statement The back row of a cinema has 12 seats, all of which are empty. A group of 8 people including Mary and Francis, sit in this row. Find the number of ways they can sit in these 12 seats if a) There are no restrictions b) Mary and France's do not sit in seats which are next to...
  23. Suyash Singh

    B What is the total number of combinations of letters in the word MISSISSIPPI?

    The total number of different combinations of one or more letters which can be made from the letters of the word MISSISSIPPI is? First i don't understand what the question means and second my answer is completely different from that in my book My working- since there are 11 letter 4 I 4 S 2 P...
  24. opus

    Order of permutations in transformations of functions

    Homework Statement This is for College Algebra. Describe what is happening to the graph of the function ##f\left(x\right)=\sqrt {1 - x}+2##. Homework Equations N/A The Attempt at a Solution This can be rewritten as ##f\left(x\right)=\sqrt {-x + 1}+2## My conundrum: Both my text, as well as...
  25. F

    Inverse of a Cycle: Proof of Inverse Cycles and Powers

    Homework Statement In the following problems, let ##\alpha## be a cycle of length s, say ##\alpha = (a_1a_2 ... a_s)## 3)Find the inverse of ##\alpha## and show that ##\alpha^{-1} = \alpha^{s-1}## Homework Equations I've observed in the previous problem that there are ##s## distinct powers of...
  26. thebosonbreaker

    B Arranging blocks so that they fit together

    I have attached an image showing a (what I believe to be) simple problem involving arranging four blocks, each of different dimensions. Yes, the blocks fit together perfectly in the first arrangement shown in the diagram when there are no gaps. I'm convinced that the solution is likely easily...
  27. T

    MHB What are the different ways to place two objects in five slots?

    I'm getting these concepts confused. If I have an object called $x$, and I have five places or slots to put the object, how many ways could 2 $x$s be places in the 5 spaces? Example: x x _ _ _ x _ x _ _ x _ _ x _ x _ _ _ x _ x x _ _ _ x _ x _ _ x _ _ x _ _ x x _ _ _ x _ x _ _ _ x x So in...
  28. B

    Permutations and Transpositions

    Homework Statement Attached are some screen shots of portion of the textbook I'm currently working through: Homework EquationsThe Attempt at a Solution My first question, why exactly can't ##\Delta## contains ##x_p - x_q## only once (note, switched from ##i,j## to ##p,q##)? As you can see...
  29. Mr Davis 97

    Showing that dihedral 4 is isomorphic to subgroup of permutations

    Homework Statement D4 acts on the vertices of the square. Labeling them counterclockwise starting from the top left as 1, 2, 3, 4, find the corresponding homomorphism to S4. Homework EquationsThe Attempt at a Solution I am not completely sure what the question is asking. It's pretty clear to...
  30. J

    Permutations of taking 2 letters from (a, a, b, c)

    Homework Statement All permutations made with letters (a, a, b, c) taken two at a time are: Homework Equations Permutation is nPr where n is total number of things, r is number of things taken at a time. If P1 objects are identical, P2 objects are identical such that P1 + P2 = n then...
  31. P

    Permutations with possibly repeating letters

    Homework Statement Find the number of distinct three letter words obtainable from the letters A,B,C,D,E in which E may occur 0, 1 or 2 times but the rest may occur only once. Homework Equations Number of combinations when picking r objects from n possibly objects in which one object is repeated...
  32. B

    Two Conjectures about Permutations in ##S_n##

    Homework Statement I have two conjectures whose truth I am interested in. The first is, "If ##\tau, \sigma \in S_n## are disjoint, then ##\tau \sigma = e## if and only if ##\tau = e## and ##\sigma = e##., where ##e## denotes the identity permutation." The second is, "If ##\tau## and ##\sigma##...
  33. MarkFL

    MHB Symmetry Groups of Cube & Tetrahedron: Orthogonal Matrices & Permutations

    This question was originally posted by ElConquistador, but in my haste I mistakenly deleted it as a duplicate. My apologies... For part (a) we can define two cyclic subgroups of order $2$, both normal, $\langle J\rangle$ and $\langle K\rangle$ such that $V=\langle J\rangle \langle K\rangle$...
  34. Kara386

    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. NatFex

    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. L

    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. Shafia Zahin

    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. M

    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. H

    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. squelch

    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. P

    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. C

    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. Nader AbdlGhani

    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. Crazorin

    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. Odious Suspect

    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. D

    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. Mark44

    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. toforfiltum

    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. K

    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. diracdelta

    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...