For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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3answers
721 views

How to count permutations with restrictions on how items are grouped

I am trying to solve the following problem: A town contains $4$ people who repair televisions. If $4$ sets break down, what is the probability that exactly $i$ of the repairers are called? Solve ...
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2answers
221 views

Counting Methods: Restricted Permutations

I have been scratching my head for a long time. The question is: How many words can be formed using all letters in the word EXAMINATION in such a way that the first two letters are different ...
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1answer
173 views

Why is $PGL_2(5)\cong S_5$?

Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
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0answers
58 views

Continuous random sampling with replacement.

Construct a set $s\subseteq[0,1]$ by sampling points in $[0,1]$ with uniform probability density $x\leq1$ so that $|s|=x$. Interpret this as a sampling frame during which data is captured. Now, ...
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1answer
28 views

Plausible gene sequences

I'm not looking for a specific answer to a question (below). I think it is likely that the 'kind' of problem I have has been studied (and has a name ;). But I don't know what that might be. So I'm ...
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2answers
757 views

How many ways are there to encode the 26-letter English Alphabet into 8-bit binary words?

I know that I need 5 bits to represent a character. All the combinations to encode the 26-letter alphabet will be 2^5? How about the 3 bits that remains from 8 bits?
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0answers
39 views

Notation for Restriction of Permutation

Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a ...
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1answer
65 views

finding out team number

A supervisor has to select a three-member project team from among her 12 employees. Unfortunately, two of the employees cannot work together on the same team. With this restriction, how many different ...
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1answer
28 views

Probability of a subset occuring within a section of a permutation

I have a list of permutations of a vector of length 24. In each position of the vector is a number between 1 and 24 and repeats are not allowed. An online tool tells me that this will give ...
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2answers
110 views

Determine no of combinations for cutting stock algorithm

I have to buy $n$ wooden logs of size 2000 each, from which I have to cut different pieces of smaller size say: 255*10 750*7 550*13 In a manner that cutting will ...
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1answer
67 views

decomposition of m-cycle in m-1 transpositions

I am searching for a proof. Every m-cycle $\sigma = (x_1 x_2 ... x_m)$ can be expressed as an composition of m-1 transpositions. I found many formulas, for example: $\sigma = (x_1 x_2)(x_2 x_3) ... ...
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3answers
1k views

Distributing identical objects to identical boxes

We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
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1answer
75 views

Find a lower bound

Let $M$ be an $N\times N$ symmetric real matrix, and let $J$ be a permutation of the integers from 1 to $N$, with the following properties: $J:\{1,...,N\}\rightarrow\{1,...,N\}$ is one-to-one. $J$ ...
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1answer
99 views

Is there a name for this type of permutation?

Let $J$ be a permutation of the first $N$ integers (1, 2, ..., $N$), so that the permuted sequence reads $(J(1),J(2),...,J(N))$. The function $J$ must of course be a bijection. Additionally, suppose ...
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2answers
162 views

problem in permutation question.

Find the number of hexadecimal numbers containing at maximum 16 hexadecimal digits with all of the digits 0,1, and A present at least once? Give your answer as a hexadecimal number.
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2answers
36 views

Number of Strings with two specific letters

How many ways can you construct a string four letters (from 26 alphabet characters) that have both the letters j and k in them?
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1answer
140 views

Permutation Formula

I am having difficulty with one minuscule detail of the permutation formula: $$n(n-1)(n-2)\cdots(n-r+1)$$ I understand that if we proceed with an $r$-permutation, then we have $r$ amount of slots, ...
4
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1answer
172 views

Dihedral group as a matrix group

I wish to consider the dihedral group as a matrix group. One way to do that is to consider it as a finite subgroup of $O_2$, a group of orthogonal $2\times 2$ matrices, defined by ...
3
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3answers
243 views

Discrete Math, anagram combinatorics

Find the number of anagrams for the word "ALIVE" so that the letter "A" is before the letter "E" or the letter "E" is before the letter "I". By before we mean any letter previous, not just immediately ...
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1answer
1k views

Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
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2answers
100 views

Basic question on permutations

I'm currently studying multilinear algebra and I felt the need to see just the basics of permutations since it's used to study symmetric and antisymmetric tensors. My doubt was: given some permutation ...
2
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3answers
74 views

Proof of Group Theory Relation - Fixed Point Sets

I was wondering if anyone could help me with a proof of the following Theorem. It is merely listed as a statement in my book... Let $A$ and $C$ be finite sets, and let $G$ be a group of ...
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3answers
107 views

Is every pure set of permutations a group?

Let $\mathcal{P}$ be the set of permutations over a finite set $\mathcal{S}$, with $|\mathcal{P}|$=$|\mathcal{S}|!$ $(\mathcal{P},\circ)$ is a finite group, where $\circ$ is composition. A subset ...
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2answers
413 views

Odd or even permutation with matrices

I know that the number of transpositions would determine the parity of a permutation like: A = (1,2,3,4,5) = (1,5),(1,4),(1,3),(1,2) = even But how would that apply to a matrix? Example: 1 2 ...
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5answers
266 views

Permutations of Symmetric Group of Order 3

Find an example, in the group $S_3$ of permutations of $\{1,2,3\}$, of elements $x,y\in S_3$ for which $x^2 = e = y^2$ but for which $(xy)^4$ $\not=$ e.
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5answers
175 views

Prove that sgn$(\sigma \tau) = $sgn$(\sigma)$sgn$(\tau)$

Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$. I think the two thing's I'm trying to show are: If sign$(\sigma)$ = sign$(\tau) = \pm 1 ...
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2answers
225 views

Inclusion & Exclusion: In how many permutations of the digits $0,…,9$ there's no continuity of 7 digits or more?

In how many permutations of the digits $0,...,9$ there's no continuity of 7 digits or more? (Ex. the number 203456789 1 should not be counted) I believe that the basic case, for the inclusion ...
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1answer
70 views

Calculating the sign of the generalised permutation

What is the sign of the following permutation. Prove your answer: $$\pmatrix{1 & 2 & \cdots&p&p+1&\cdots & \cdots &p+q \\ q+1 & \cdots & \cdots & q + ...
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1answer
147 views

Bit permutations and collisions of compression function

I'm having trouble finding a good method for solving the following problem: If $n$ is a positive integer, let $S_n$ denote the group of permutations of the set $\{1,2,\dots, n\}$. For a permutation ...
4
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3answers
128 views

Help finding all elements of order 2 in $S_6$.

I am trying to find all elements of order 2 in $S_6$. I am trying to understand how to achieve this. Here is my attempt. We need only count the number of permutation of the forms $ (a_1 a_2)\\ ...
0
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1answer
41 views

Listing the elements of $A(3)$

List the elements of $A(3)$ and give the order of each of them. This is about permutations in number theory ... to clarify that $A(n)$ Thanks!
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2answers
109 views

Finding the sign of each permutations

How to find the sign of each of the following permutations? 1, (1 2 3 4 5)(8 7 6)(10 11) 2, (1 3 5 7 9 11)(2 4 6 8 10) 3, (1 2)(3 4)(5 6 7 8)(9 10) 4, (1 2 3 4 5 6 7 8)(1 8 7 6 5 4 3 2) Help ...
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1answer
119 views

Explanation on step $\rho$ of the SHA-3 algorithm

I'm working on implementing SHA-3 in a PIC microcontroller. In the block permutation, I don't quite understand step $\rho$: Bitwise rotate each of the 25 words by a different triangular number 0, ...
2
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1answer
61 views

Any comprehensive material to revise the mathematics

I left school long back and so my mathematics knowledge also fades out. I am trying hard to re-collect the basics about log / permutaion / combination / probability / polynomial equations. I tried ...
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3answers
30 views

this can be solved bypermutation and combination based problem

How many three are there whose hundred digit is greater than tens digit which in turn is greater than the unit digit? Ans:I tried it But couldn't solve..
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2answers
68 views

Different actions of an affine primitive group?

Fairly new to group actions and I'm having trouble finding answers to these in textbooks... Say we have a primitive action of $G$ on $\Omega$, with regular elementary abelian socle $N$. Now suppose ...
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1answer
238 views

How do combinations (not permutations) relate to group theory?

First question. I'm just generally curious about combinations in group theory. How do they relate? If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
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2answers
78 views

Why such a representation is said to be unique when it's known that disjoint cycles commute?

My question is about the underlined statement of Herstein, Topics in Algebra (2nd Ed.) Why such a representation is said to be unique when it's known that disjoint cycles commute? Also ...
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4answers
7k views

Odd/Even Permutations

How do you classify a permutation as odd or even (composition of an odd or even number of transpositions)? I somewhat understand the textbook definition of it but im having hard time conceptualizing ...
0
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1answer
175 views

Permuation with repeated letters and consecutive letters not same

I have been trying to solve a question on permutation and haven't really been successful. I want to generate all the permutations of a specified length that start with a letter and end with the ...
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2answers
178 views

Other proofs that subgroups of $A_5$ have order at most 12

How can it be proved that any subgroup of $A_5$ has order at most 12? This is [Herstein, Problem 2.10.15], which also gives the hint that I can assume the result of the previous problem that $A_5$ ...
3
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4answers
108 views

Is my textbook wrong?

My textbook says (without explaining how it is done): $$\begin{pmatrix} 1\ 2\ 3\ 4\\ 2\ 1\ 4\ 3 \end{pmatrix}\begin{pmatrix} 1\ 2\ 3\ 4\\ 2\ 3\ 4\ 1 \end{pmatrix}=\begin{pmatrix} 1\ ...
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2answers
283 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
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1answer
71 views

number of ways of placing balls on plate

There are n plates places in a line and unlimited number of red balls with values from 1 to ...
1
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1answer
438 views

Doubt on married couple seating arrangement problem

I am going through a solution of the following problem. "How many ways there are there to seat $n$ couples around a circular table with $2n$ chairs such that no couple sits next to each other, i.e., ...
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1answer
67 views

Number of arrangements around a table

My doubt is based on two observations : 1) On top of a round table (which is rotatable) there are $n$ places to sit and we need to place $n$ people. How many ways is it possible to permute them ? ...
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0answers
45 views

Classification of $n \times n$ matrix in which each two components differ in each row and column up to automorphism

Suppose, we define a class $A$ of $n \times n$ matrix as follows: $$\text{In each Row }i, \text{for any} j,k\ (1 \leq j,k \leq n )\ a_{ij} \neq a_{ik} $$ $$\text{In each Column }l, \text{for any } ...
4
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1answer
182 views

Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
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0answers
48 views

Cycle of length

I'm learning permutations and came upon this question which made me freeze. So to say it in my own words, it asks that how many permutations in $S_n$ do not have a cycle of length one in their ...
3
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1answer
180 views

Orders of centralizers $C_G(g)$ in a group of order 60?

Given a group $G$ of order 60 with 24 elements of order 5, 20 of order 3, and 15 of order 2, how do we find the sizes of centralisers of elements of $G$ without proving $G\simeq A_5$? By considering ...