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

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5 green balls, 4 blue balls, 3 red balls. Number of combination of balls, which can be chosen taking at least one green and blue ball? [on hold]

So at least we have to take 2 balls Answer would be sum of selections of 2 balls + 3 balls + 4 balls ....+ 12 balls... for 2 balls total selection 1 (1 green, 1 blue)... for 3 balls total selection 3 ...
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2answers
39 views

Geometric Interpretation of S3

My impression was that the symmetric group $S_3$ acts on the vertices of a labeled triangle. However, I am not sure this is the case anymore, because of the following. (The triangle is labeled as ...
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1answer
19 views

Counting the number of equivalent words (having the same letters, with the same order of vowels)

2 words are considered to be equivalent of they have the same order of vowels and the same alphabets and the same number of alphabets as in the original word. Find the number of equivalent words of ...
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2answers
29 views

Probability involving chess board

if 2 cells are chosen at random on a chess board what is the probability that they will have a common side i tried solving the question by considering different cases for the cells on: 1. corner 2. ...
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1answer
21 views

How to find the number of permutations with offset restriction

First question. Okay I have this problem that I've been trying to figure out for a while. I'm writing a computer program I need to quickly calculate the permutations of a set with 'n' elements with a ...
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1answer
24 views

Circular permutation, sitting 6 people in a round table

6 people sit down a round table. 4 of them belong to group X and 2 of them belong to group Y. How many ways are there for the 6 people to sit down by taking into account that the 2 people in group Y ...
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1answer
47 views

How to find number of subsets

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$
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21 views

Sign of Composition of Permutations

Let $\sigma$ be a permutation in $S_5$ with a sign of $-1$. Let $\pi$ be any other arbitrary permutation of $\{1,...,5\}$. What is the sign of the composition $\pi^{-1} \sigma \pi$? Is there any ...
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18 views

Prove $(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$

Show that $$(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$$ I have been trying to show this for the past few ...
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35 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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1answer
9 views

How to figure out how many possible sequences contain a specific criteria

If a 6-sided die is rolled 5 times and each roll is recorded as an element of set A (|A| will be 5 after all rolls), How many results out of all the possible results will have exactly two 4's as ...
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1answer
29 views

Probability related question, Permutations, combinations [on hold]

Im doing a practice problem for an upcoming test, I had a hard time figuring out this question, could anyone walk me through it?
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1answer
36 views

$5$ green dyes, $4$ blue dyes, $3$ red dyes. Number of combination of dyes, which can be chosen taking at least one green and blue dye? [on hold]

$5$ different green dyes, $4$ different blue dyes, $3$ different red dyes. The number of combination of dyes, which can be chosen taking at least one green and blue dye ?
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0answers
24 views

Monotone subsequence in a random permutation

I wish to compute the probability of having a log(n) length consecutive monotone subsequence in a random permutation of {1,...,n} (log with base 2). I'm trying to show it's $\leq1/n$, does it make ...
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2answers
38 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
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1answer
26 views

$r$-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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4answers
38 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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23 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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1answer
33 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
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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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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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
24 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
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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30 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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37 views

Probability of getting 6 letters right [duplicate]

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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2answers
450 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. [closed]

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
29 views

How to prove this result using Permutations? [closed]

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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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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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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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 ...
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1answer
73 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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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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3answers
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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
44 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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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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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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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 ...
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1answer
35 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
39 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
32 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
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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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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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 ...
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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 ...