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

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r-cycle to a power k is also an r-cycle if and only if gcd(k, r) = 1

Let $\sigma$ be an r-cycle in $S_n$ and let k$\in\Bbb Z$. Show that $\sigma^k$ is also an r-cycle if and only if gcd(k,r)=1.
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3answers
20 views

Permutations - selection

Give the total number of possible arrangements of 3 letters chosen from the word CALCULUS. The answer is 96, but all I can get is 5P3=60 (permutations of 3 from 5 different elements), or 8P3 adjusted ...
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22 views

Filling a 5x5 array with X-s and O-s

Consider a 5x5 array. In how many ways can we fill the array with X-s annd O-s so that no two consecutive rows are identical? My tutor gave us the following answer: 2^(25) - [4*2^(20) - 6*2^(15) + ...
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2answers
212 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [1]

Let $A$ be an $n\times n$ matrix with entries in the set $\{0,1\}$ which has exactly two ones in each column and two ones in each row. Give necessary and sufficient conditions for the rank of $A$ to ...
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1answer
85 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [3] [duplicate]

Let $A$ be an $n\times n$ matrix with entries in the set $\{0, 1\}$ which has exactly two ones in each column and in each row. Give necessary and sufficient conditions for the rank of $A$ to be $n$.
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50 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [2] [duplicate]

Let $A$ be a $n \times n$ matrix with entries on the set $\{0,1\}$, with exactly two ones on each column and two ones on each row. Give necessary and sufficient conditions for rank$(A)$ to be $n$. I ...
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1answer
31 views

A variation of a combination and a permutation, I think?

The scenario is that 6 people have the option of choosing 8 doors and we want to know each door a person goes through. I have four/five questions based on this. 1) How many different ways can 6 ...
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2answers
15 views

Probability of a word where 2 letters do not follow each other

I have seven letters, say A, B, C, D, E, E, G. I have figured out how many distinct possible combinations I can have as $7!/2!$. My question is, how many of these will have the two E's separated? I ...
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1answer
45 views

Find orbit of $1$ for $\sigma$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ $ 1 \mathop{\rightarrow}^{\sigma} 3 \mathop{\rightarrow}^{\sigma} 4 ...
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1answer
16 views

permutations of objects containing non distiguishable objects in sample

if we have 3 types of objects A,B,C . If I want to make permutation without repeat n! = 3! = 6 but if i will take r sample so ...
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1answer
43 views

Combinatorics: Prove the number of matches in a Singles Tournament

I was working my way through some problems in Discrete Maths by Rosen, when I came across the following question: There are x players in a singles badminton tournament Show that there are ...
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34 views

give a group that is isomorphic to the figure.

I think if I get help with one of these I should be good on the rest. 23) is concerned with figure a) It has one symmetry and 4 possible points. seems like it would have two elements that map to ...
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0answers
15 views

4 lone women, 7 roses and 5 “Different” delicious cookies. [on hold]

In how many ways can we distribute 7 identical roses and 5 different cookies among 4 ladies so that no lady gets more than 3 roses? I've been thinking about this exercise for over an hour now.. ...
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1answer
20 views

Permutation question, 9 seats.. 3 nationalities.

There are 9 seats in a row, 3 Chinese people.. 3 Russians and 3 Poles. How many ways are there for those people to be seated, so that they don't sit next to a person of the same nationality. Would ...
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1answer
18 views

How many undirected graphs are possible with $4$ labelled vertices such that exactly $1$ edge is present?

I have drawn the graph and the result is $6$ graphs are possible. A simple graph can have a maximum of $\Large\binom{n}{2}$ edges and each edge can exist or not exist. Therefore, ...
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2answers
36 views

no. of all ordered tuples (x,y,z) such that x,y,z are all positive integers that satisfy the equation x + 2y + 3z = 30? [on hold]

How do I find the number of all ordered tuples (x,y,z) such that x,y,z are all positive integers that satisfy the equation x + 2y + 3z = 30 ? Is there any easy and less time taking method to solve ...
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1answer
24 views

Number of subgroups and normal subgroups

I am struggling to understand how to calculate the nunmber of subgroups with permutations, for example: How many normal subgroups does S3 have? How many subgroups of order 4 has group S4? And does ...
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1answer
70 views

Number of adjacent permutations.

What is the number of permutations for any number of adjacent elements swapping places at the same time in an array of length $n$? My solution: I think that we only need to count the number of ...
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0answers
28 views

help selecting a committee with replacement [on hold]

The Penguin Society has an Wine Committee (WC) consisting of five members and a Beer Committee (BC) consisting of six members. • Assume there are n ≥ 6 members in the Penguin Society. Also, assume ...
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2answers
34 views

Probability of getting 6 letters right

A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probability that he gets exactly 6 ...
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4answers
438 views

Is it always true that $(a,b,c)(a,b,c) = (a,c,b)$?

I noticed that $(1,2,3)(1,2,3) = (1,3,2)$, and I also noticed that $(1,4,3)(1,4,3) = (1,3,4)$. Now, my question is whether or not it is true that for any permutation $(a,b,c)^2 = (a,c,b)$?
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2answers
449 views

To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit. [on hold]

I have a question and got strucked on this.. To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit. Please guide me ...
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1answer
28 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.
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How to combine possible permutations of two sets to find number of combined permutations

I hope the title accurately describes the question. I have a question that asks: There are 7 male swimmers and 5 female swimmers. If there is a gold, silver, and bronze medalist male swimmer, and a ...
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3answers
58 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
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2answers
28 views

Histogram of duplication in n choose k

Imagine having 17 balls to distribute to 4 people. One algorithm for distributing these balls is to give each ball to one out of the four randomly. This means, in an extreme case, it is possible for 1 ...
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3answers
33 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
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1answer
31 views

Easiest way of finding a root of permutation?

I've been searching extensively for the simplest way of finding a root of a permutation, but I can't understand half of the things that I've found. Let's say we have 2 permutations: $\alpha^2 = ...
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257 views
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Calculate cardinality of 8-digit strings composed of zeros and ones

How can i calculate cardinality of a set made of 8-digit strings composed of zeros and ones? In general, assuming that digits can repeats. My attempt Let $D$ be the domain composed only by one or ...
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1answer
27 views

permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. ...
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1answer
42 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
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1answer
33 views

cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
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1answer
45 views

How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?

What I tried was: (9P4)/3!*2! This gave me a wrong answer (since the answer is 626). I'm unable to make use of the hint provided in my book: "make cases". Any help would be appreciated. :)
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1answer
51 views

Describe the subgroup $K\leq S_4$ of order 8

How do I construct the subgroup $K$ (a subgroup of $S_4$ of order $8$) ?
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Is there a name for these oscillations in the self-similarity of a set under the action of a cyclic group?

I don't know much about group theory and card-shuffling theory, so this may already have a name I don't know about. I often shuffle a deck of cards using a method that is defined by a particular ...
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2answers
92 views

Number of lists at some Kendall-Tau distance

Given a ranked list (permutation) $R$ of $n$ elements, how many permutations of the same elements are there at Kendall-Tau distance $d$ from $R$ $(0 \le d \le \frac{n(n-1)}{2})$? Example: If $R = ...
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0answers
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Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
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1answer
30 views

Ways of writing an $n$-cycle as product of a $2$-cycle and $n-1$ cycle.

We know that any $n$ cycle can be written as a product of a $2$-cycle and an $n-1$ cycle; but this decomposition is not unique: $(123)=(12)(23)$ and $(123)=(23)(31)$ [product taken from right to left ...
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1answer
393 views

Number of permutations with a single fixed point

I know that the number of permutations with no fixed points over a set with $n$ elements approaches $\frac{n!}e$ as $n$ grows. I'm interested in finding a limit (if there's exist) for the number of ...
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0answers
25 views

Frobenius groups of order 36 [closed]

Is there a Frobenius group of order 36? If yes, what is it's structure as semidirect product of two subgroups?
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1answer
38 views

Bringing a permutation back to the identity

I'm working with transposition distance (nothing to do with algebraic transpositions) on given permutations. Given a permutation, how many moves (transpositions) will it take to get back to the ...
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2answers
31 views

Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) [duplicate]

Find Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) I think we need to find order of centralizer of given permutation but how to find it?
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1answer
58 views

Number of ways to pick N numbers from 0,1,…,N-1, with possible duplication, with sum equal 0 mod N

We have the numbers $0,1,2,....,N-1$ in $\mathbb Z_N.$ I want to pick $N$ numbers from these. These are the rules: Duplication may occur We don't care about ordering, $00041$ is equivalent to ...
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1answer
30 views

With $m>n$ , In how many ways $m$ men and $n$ women can seat in row for a photograph so that no two women are adjacent? [duplicate]

Given $m>n$ , In how many ways $ m$ men and $n$ women can seat in row for a photograph so that no two women are adjacent? My effort : There are $m-1$ gaps if $m$ men are seated. Now we have to ...
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2answers
31 views

To calculate no of substring in length of 12 string

How many bit string of length 12 contains 01 as a substring ? I arrived at 2^10 . taking 01 as one set and remaining as other set
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1answer
105 views

Factorizing elements of a group into a product of generators.

$$ s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n) $$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
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1answer
36 views

Linear Permutations of $n$ objects

Suppose there are $n$ distinct objects $O_{1},O_{2},O_{3},\ldots,O_{n-1},O_{n}$. We have to find out the number of ways we can arrange them. But, there is a catch. We have to arrange them such that ...
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1answer
36 views

Nth pemutation of Lexicographic String

Can someone please explain the logic behind the mathematical equation, that for finding the Nth Lexicographic rank of a string the Leading Entry is $a_q$ if $k=q\cdot (n!)+r.$ The link to the problem ...
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0answers
19 views

Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...