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

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30 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?

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 ...
2
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0answers
31 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 ...
0
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1answer
20 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 ...
0
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1answer
68 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 ...
0
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0answers
19 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 ...
0
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2answers
29 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 ...
4
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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 $(a,b,c)^2 = (a,c,b)$?
2
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2answers
434 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 ...
2
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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.
2
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0answers
18 views

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 ...
0
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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 ...
0
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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 ...
1
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3answers
30 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 ...
1
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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 = ...
2
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0answers
253 views
1
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0answers
13 views

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 ...
1
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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. ...
0
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1answer
40 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?
1
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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 ...
1
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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. :)
3
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1answer
48 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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0answers
8 views

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 ...
1
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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
32 views

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 ...
0
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1answer
28 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 ...
2
votes
1answer
391 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 ...
0
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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?
3
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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?
6
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1answer
57 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 ...
0
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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 ...
0
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2answers
28 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
3
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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 ...
0
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1answer
35 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 ...
1
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1answer
34 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 ...
2
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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 ...
1
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2answers
44 views

Number of circular permutation of word 'CIRCULAR' [closed]

Hey please help me with this question... Find the number of circular permutation of the word 'CIRCULAR'. Number of circular permutaion is (n-1)!
2
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2answers
46 views

Letters of the word “PARAMETER” [closed]

I have one question that bothers me. The total number of words that can be made by writing the letters of the word PARAMETER so that no vowel is between two consonants. The answer is 1800. I couldn't ...
1
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0answers
25 views

In how many ways can I move $M$ steps such that I do not leave the $N$-dimensional space at any point?

Suppose that currently I am at some position $(p_1,p_2,p_3 \dots p_N)$ in an $N$-dimensional space. The dimensions of the space is $(d_1,d_2, \dots d_N)$. In a step, I can walk one step ahead or ...
4
votes
1answer
436 views

Permutations of a set with a conditional subset

Using the digits 1, 2, 3, 5, 6, 8, 0 only once, how many 4-digit numbers could be constructed if the number is even? This is an exercise from an online course I'm taking. The given solution suggests ...
-1
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1answer
39 views

Number of $4\times K$ board arrangements such that there exist no square of size 2 having all black cells

Suppose a person has a board of size 4 x K and each cell of this board can either be black or white. The person and his girlfriend come to a conclusion that they don’t like black squares. We decide to ...
0
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2answers
14 views

Related question permuatation and combination 1

How to identify the question is permuatation or combination? And below is some question: i cannot solve. Show how to solve it. Thank you.
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1answer
22 views

probablity - number of const points in permutation

$\pi$ - random permutation. 1. Compute expected value of const points of $\pi$ 2. Compute Variety. 3. Estimate the probability that $\pi$ has more than $n/2$ const points Firstly, I have a problem ...
0
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1answer
63 views

In $S_n$, if $ε = α_1α_2 \cdots α_r$ where $α_i$ is a $2$-cycle, then $r$ is even.

In $S_n$, if $ε = α_1α_2 \cdots α_r$ where $α_i$ is a $2$-cycle, then $r$ is even. I don't know how to start. Note, $ε$ is the identity of the permutation group $S_n$.
5
votes
5answers
782 views

How many Arrangement of “AMAZED” letter E Positioned between two A's (Not necessarily Flanked)

I considered 'AEA' as one letter so there are 4 letters which can be arranged in 4!=24 ways. But my sheet is telling its 120 How? Please HELP! & What is flanked meaning here?
1
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3answers
35 views

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd , a homomorphism?

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd an injective homomorphism ? I can show that if it is a homomorphism ...
0
votes
1answer
19 views

Permutation: How many numbers of n digits are possible for which product of its digits is a perfect square.

I need to find total numbers from 1 to 10000 whose product of digit is a perfect square. eg: 49 (4*9=36), 236 ( 2*3*6=36) etc. Till now i have figured out these things: 1) For a number to be a ...
3
votes
2answers
33 views

Cycle structure in a symmetric group

I have a bit of a problem. I'm currently reading about permutations, and I have a little exercise that asked me to find all cycle structures in $S_6$. I came up with the following $ ( -)\\ (- -)\\ (- ...
2
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2answers
37 views

Problem on circular permutation

In how many ways can $x$ people be seated at a round table so that all will not have the same neighbours in any two arrangements?
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3answers
129 views

Seating permutations for 10 people where 2 people always sit together and 2 people never sit together

We have to seat 10 people in a row. Condition: two people always sit together and two people never sit together. My attempt: Let the two people who always sit together be taken as 1 person for the ...