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

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140 views

Orbit and Stabilizer

Are the following definitions essentially the same: Orbit: Let $G$ be a group of permutations of a set $S$. For each $s \in S$, let $\operatorname{orb}_G(s)= \{f(s) \mid f \in G\}$. The set ...
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1answer
69 views

Multiplying Cycle Permuations

I am having trouble multiplying permutations in cycle notation. (1 3 4 5) (2 3 4) = (1 3 5) (2 4) I do not understand how this product is determined. My answer is (1 3 4) (2 5). I have come to this ...
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2answers
98 views

Number of combinations and permutations of length $5$ obtained from “Mississippi”

Please help me to solve this: How many permutation & combination can be be formed from the word $MISSISSIPPI$ taking $5$ at a time?
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1answer
567 views

4 Element abelian subgroup of S5.

I have a homework question from my intro to group theory class. Question: Find a 4 element abelian subgroup of $S_5$. Write it's table. This is where I've gotten so far, but I don't even know if ...
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1answer
395 views

Prove why this algorithm to compute all list permutations works

Note: this is not homework or for a class, as I'm no longer in school. Let's say I have a list of characters {1,2,...,N} and I want to generate all permutations. For example, if I had {1,2,3} the ...
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2answers
157 views

How many triangles can be made with those holes?

A 5 by 5 square lattice is formed by drilling holes in a piece of wood. Three pegs are placed in this lattice at random. Find the probability that three randomly chosen points of a 5 by 5 lattice ...
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5answers
2k views

Find all permutations in increasing order

Given a set of distinct numbers, say, {1, 2, 3, 4, 5, 6}, find all permutations containing 3 numbers. All the permutations have to be in ascending order. For e.g., some correct permutations would be ...
2
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2answers
267 views

Which elements of $S_8$ are in the subgroup of rigid motions of a cube?

Let the set $S\colon= \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$. Then which permutations of $S$ will appear in the group of rigid motions of a cube, which is a subgroup of $S_8$, the symmetric group on 8 letters? ...
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1answer
444 views

How to find the subgroups of S4 generated by these sets.

How do I find the subgroups of $S4$ generated by these sets in each case? $A = {(1,3),(1,2,3,4)}$ $B = {(1,2,4),(2,3,4)}$ $C = {(1,2),(1,3),(1,4)}$
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1answer
73 views

Ball selection combinatorics problem

Say I have 9 uniquely colored balls. I want to select them into 3 different groups. Each grouping must contain at least 1 ball, and all the balls must be selected. The first 2 groupings are ...
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1answer
58 views

Find The Number Equation Solutions

Find the number of non-negative integer solution of the equation: $$5x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=14$$
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1answer
117 views

Generators of permutation group

I want to proof that $S_n$ is generated by the set of transpositions ${(1,2),(1,3), \ldots , (1,n)}$ using that $(k,j) = (1,k)(1,j)(1,j)$ but I don't know how to continue. I know this is a easy ...
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3answers
267 views

Permutation question: Four seats for 14 people, how many ways?

A club has 6 female and 8 male members. A president, VP, secretary and treasurer. In how many ways is this possible if?... a) an equal number of men and women hold office? b) the president or VP is ...
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2answers
137 views

Maximum number of seating plans

15 people will be seat in a row of 15 chairs. Two seating plan are considered the same if two plans share same adjacent quadruples. What is the maximum number of seating plans can be made? For ...
0
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1answer
256 views

Permutations and symmetric groups

Suppose that a permutation $f$ is the product of disjoint cycles $f_1,f_2,\dots, f_m$. Show that $o(f)$ is the least common multiple of $o(f_1), o(f_2),\dots, o(f_m)$. Really lost with the question.. ...
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0answers
12 views

Order of a conjugate permutation [duplicate]

From Finan: Let $\sigma\in{S_n}$, define the $\mathbf{order}$ of $\sigma$ to be the smallest positive integer such that $\sigma^m=(1)$. Prove that if $\sigma$ has order $m$ then ...
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1answer
204 views

Find number of ways of selection of one or more letters from the word: AAAABBCCCDEF

Hello the question is above , I can't understand such type of questions, please help. the answer is 479 but how does it come?
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1answer
60 views

Can this polynomial transformation produce new symmetry?

I've got a polynomial transformation on $\mathbb{R}^6$, and I have a conjecture about it, but I'm having a hard time proving it. The transformation looks like this: $ u:= abcde + abc + abe + ade + ...
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1answer
1k views

Distributing $n$ different things among $r$ distinct groups such that all of them must get atleast $1$

In how many ways can we arrange $7$ different things to $3$ people, such that all of them must get at least one? We know that if we have $n$ identical items which will be distributed in $r$ distinct ...
2
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0answers
90 views

Number of permutations possible?

Given two permutation of $1, \ldots, N$. Where 3<=N<=1000 Example For $N=4$ First is $\begin{pmatrix}3& 1& 2& 4\end{pmatrix}$. Second is $\begin{pmatrix}2& 4& 1& ...
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2answers
48 views

Finding a set of a subgroup in Sx

I cannot think of a set $X$ and a subgroup $H$ of $S_x$ which is isomorphic to $\mathbb{Z}$.
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0answers
208 views

Magic Square Combinatorics

This question has been noted to be close to a Project Euler question. Please Help me with this question:Considering a 4*4 magic square ,How many ways are there to fill each square with an integer ...
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2answers
305 views

Probability with combinations of 1-77 numbers and alphabets

(Sorry about me not being able to explain this problem with perfect mathematical terms) Consider the following data set its a combination set of 1-77 ...
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1answer
68 views

Communicating with Language

Can you please help me with this problem? There are $n$ people living on a planet. It is known that their planet has $6$ languages and each person knows every language. It is also known that any two ...
0
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1answer
62 views

Permutation Question of Dihedral Group Order

Let $n \ge 3$. Number the vertices of a regular n-gon by the numbers $1,2,...,n$. Each symmetry of the n-gon corresponds to a permutation of its vertices and hence to an element of $S_n$. The subgroup ...
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1answer
105 views

product of disjoint cycles

Define a permutation $\sigma \in S_7$ by letting $\sigma (1) = 3, \sigma(2)=2, \sigma(3) = 7, \sigma(4)=5, \sigma(6)=1, \sigma (7)=6$. I need help with presenting $\sigma$ as a product of disjoint ...
0
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1answer
73 views

A subgroup of $S_n$ of odd order is contained in $A_n$

I saw this question. The original questioner asserted that the Lagrange's Theorem is sufficient to solve the problem, but I think that the theorem does just say that the order of $H$ divides the order ...
0
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3answers
296 views

Decomposition of a cycle as a product of transpositions

Can someone please explain the rules pertaining to different ways to write a cycle decomposition as products of 2-cycles, an example from textbook: I understand this $$ (12345) = (54)(53)(52)(51) ...
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1answer
133 views

Finding the smallest set on which a group acts faithfully

Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the ...
0
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1answer
96 views

Predicting the Nth combination in sequence of A and B

First I need a way of generating every possible combination of As and Bs in an array of 2048 items. Once I have that would it be possible to row N without action generating every row between 0 and N? ...
2
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1answer
65 views

Signs of products of permutations with given values sums

There is a funny property of permutations, which is valid for $n=2,3,4$, but it would be interesting to know if it is a general fact. Let $\sigma_1,\sigma_2,\sigma_3$ be three permutations of numbers ...
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1answer
88 views

Permutations with repetition for some elements

Suppose we have $N$ slots, each of which can be filled with $X$ options, but $2$ of these slots can only be filled in $1$ way (out of $X$ ways), then what is the number of permutations possible ? For ...
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1answer
125 views

Finding the permutation that shows two permutations are conjugates method?

Problem: Given $\sigma=(12)(34)$ and $\gamma=(56)(13)$ find $\tau\in S_6$ with $\tau^{-1}\sigma\tau=\gamma$ Attempt: I'm kind of new to this but from what I understanding find $\tau$ that satisfies ...
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2answers
50 views

Possible combinations of digits

What is the probability that a random $r$-digit number $(r \geq 3)$ contains at least one $0$, at least one $1$, and at least one $2$? My initial guess was $1-(\frac{7}{10})^r$ seeing that it's $1$ ...
2
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1answer
65 views

Number of permutation with non-consecutive blocks

How many strings are there consisting of exactly M A's, N B's, and K C's so that the string BC does not appear? For example, when M=3, N=1, K=1, $$ABACA$$ counts as a valid string whereas ...
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1answer
177 views

Number of ways of arranging numbers with given max difference

How many ways are the there to arrange n numbers out of m numbers (1 to m) so that the difference between the max and min numbers of those n numbers is D which is given. For example : n = 4 m = 3 ...
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2answers
467 views

Ways to select 3 members from 5 candidates

At an election there are 5 candidates and 3 members are to be selected. In how many ways a voter can vote? My attempt: 1st member can be chosen in 5 ways, 2nd in 4 and 3rd in 3. So, $5*4*3=60$. ...
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4answers
1k views

Count of 3-digit numbers with at least one digit as 9

Find the number of $3$ digit numbers (repetitions allowed) such that at least one of the digit is $9.​$ I've posted my answer below. If there is a better way to solve this question, I would be ...
0
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1answer
119 views

Every nontrivial subgroup $H$ of $S_9$ containing some odd permutation contains a transposition. [duplicate]

This is a true or false question. Apparently, it is false, but I don't follow. Clearly, if it contains an odd permutation, and an even/odd permutation is defined by the number of transpositions it ...
0
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1answer
36 views

Permutations with some fixed numbers

You have to fill 4 spaces with 3 numbers (4, 5, 6) such that the numbers 4 and 6 appear atleast once in every case. Find the number of such unique permutations. [Ans. 50] How do you go about solving ...
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0answers
45 views

Count of 3-digit numbers

How many different three digit numbers can be formed with the digit $1,2,3,4,5,6,7,8$ none of the digit being repeated in any of the numbers so formed? $120/1200/180/270$ My attempt: $8*7*6=336$, ...
0
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1answer
70 views

Minimum moves to transform a list to another?

Given two list of n positive elements. We are allowed to perform only one transformation which is to increment each element of the list except one. What are the minimum number of transformation ...
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4answers
533 views

How many ways to arrange Lego bricks on a Lego board?

Let's say I have a board like this one (though significantly smaller, it's 4x7) and I have two 2x3 bricks. I'd like to know how many ways to arrange the bricks on the board. The bricks should ...
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1answer
95 views

Select r items from a set with multiplicity k and total items n.

Let N be a set of n distinct objects having the same multiplicity k. For instance, N={1,1,2,2,3,3} where n=3 and k=2. Now I want to select r numbers from ...
0
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1answer
264 views

Writing a permutation as products of transpositions

If a can write a permutation $\sigma$ as a product like $\Delta \alpha \beta$, where $\Delta$ is a product of transpositions (in fact, anything) and $\alpha$ and $\beta$ are two disjoint ...
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1answer
2k views

How many distinct 4 letter words can be created using the 26 alphabets.

How many distinct 4 letter words can be created using the 26 alphabets. I have a project in which I can enter the name of the branch ID which is a 4 letter word that can be of any combinations. I want ...
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1answer
55 views

Can an infinite permutatation be decomposed into finite number of infinite cycles?

Let $\sigma \in Perm(\mathbb{N})$ the set of permutations on the naturals. Then can $\sigma$ be written as a finite composition of possibly infinite disjoint cycles?
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1answer
32 views

Restricted permutations revisited!

In how many ways can we arrange $n$ different things at $r$ places (each of $r$ places can have any of the $n$ things)repetition allowed,such that $2$ of the $n$ things are always included? Foe ...
2
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2answers
125 views

$(123)$ and $(132)$ are not in the same conjugacy class in $A_4$

Could you tell me how to show that $(123)$ and $(132)$ are not in the same conjugacy class in $A_4$? I know that all 3-cycles can't be in the same class, because the order of each class must divide ...
1
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1answer
43 views

Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity when $G$ is infinite?

Let $G$ be an infinite transitive permutation group acting on a set $\Omega$. Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity for $G$ in $\Omega$? $G_\alpha$ is the set of elements of ...