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

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The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
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1answer
24 views

Do disjoint cycles commute?

When a given set is finite it is clear. I'm asking the general case. Let $X$ is an arbitrary set. Let $\sigma,\tau$ be disjoint cycles on $X$. Then do they commute?
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1answer
13 views

How many way can 7 friends line up if there are certain conditions?

How many ways can 7 friends line up if Ann, Beth, and Chris have to stand next to each other where Ann is ahead of Beth and Beth is ahead of Chris? Would it simply be $5*4*3*2*1=120$ ways? Expanding ...
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1answer
22 views

What is the definition of “disjoint cycles”?

I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics. Below, i list my definitions for cycle and orbit. Let $X$ be a nonempty set. Let ...
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2answers
30 views

What is “the orbit of a permutation”? Is the term “orbit” consistent with that for group action?

reference: What is the orbit of a permutation? To be honest, i don't understand the answer in the link. The orbit of a group action is defined as follows: Let $G$ be a group acting on a set $X$. ...
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1answer
24 views

Prove that $sgn(\sigma_1 \circ \sigma_2) = sgn(\sigma_1)sgn(\sigma_2)$

Lete $n\in \mathbb{N}$. Show that the transformation $$sgn: S_n \rightarrow \{\pm 1\}$$ (where $S_n$ is the set of all permutations of the integers in the set $\{1,...,n\}$),given by $\sigma \mapsto ...
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1answer
20 views

Analog of Birkhoff's theorem for doubly stochastic matrices

Birkhoff's theorem states that extreme point of the set of doubly stochastic matrices are permutation matrices. An $n \times n$ matrix $A$ is doubly stochastic if each row and column sums to 1. What ...
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28 views
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23 views

Permutations of two photo frames

Please help with this permutations question. I'm trying to use the permutation formula to calculate it but don't know where to begin: $$\frac{n!}{(n-r)!}$$ Here's the problem: ...
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2answers
22 views

permutation with four fixed numbers [on hold]

My problem appeared to be part of permutation but not sure. I have a fixed length of 4 digits with 2 variable digits. say i have ...
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1answer
29 views

Envelopes and Mailboxes

We suppose $n$ and $p$ are two positive integers. A) In how many ways can you divide $p$ identical envelopes in $n$ mailboxes? (Each mailbox can hold several envelopes at the same time) B) In how ...
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1answer
22 views

Count no. Of ways

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.
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1answer
32 views

Probability of item distribution with a restriction

I'm having a hard time analyzing my research data, and was wondering if anyone had any suggestions? I've reworded the question so it is presented more like a statistics problem. There are $x$ number ...
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2answers
35 views

Sum of Binomial Series of form $\binom{2000}{3k-1}$

Find the Value of $$ \binom{2000}{2}+\binom{2000}{5}+\binom{2000}{8}+\cdots+\binom{2000}{1997}+\binom{2000}{2000}$$
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2answers
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(12345) is an even permutation of S_5. True or False?

The answer i had for this question was True, yet i'm not sure. Well, from what I know so far was that: $(12345)$ can be expressed as a number of 4 transpositions such as: $(12)(23)(34)(45)$ which is ...
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1answer
29 views

Conjugate subgroups of $S_4$

$A = \langle (1,2,3),(1,2)\rangle$ $B = \langle (1,2,4),(1,2)\rangle$ $C = \langle (1,3,4),(1,3)\rangle$ $D = \langle (2,3,4),(2,3)\rangle$ I want to proof that these subgroups of $S_4$ ( which ...
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3answers
79 views

How do I solve for n in this permutation question?

I have the following question: Solve for n: $$_nP_3 = 6_{n-1}P_2$$ I don't know how I should begin to tackle this problem? Any tips/help would be appreciated.
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2answers
35 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
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0answers
20 views

Fixed points and permutations.

Let $\psi ,\varphi \in {S_n}$ two permutations. Let $M$ a matrix such that $a_{i,j}=1$ iff $i=\sigma(j)$ where $\sigma \in S_n$ ($0$, otherwise) I already showed that $tr(M) = \left| {\left\{ {k \in ...
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(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
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31 views

Finding out the triangle numbers [on hold]

How will I find the number of errors without counting the triangle??? DO i have to find out the points and then permutations?
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27 views

No. of 4 digit nos. which can be formed containing at most 2 digits? [closed]

No. of 4 digit nos. which can be formed containing at most 2 digits? I'm getting $585$ as the answer. Please tell me where I'm wrong: (1.) 9 different digits - 9 nos. (2.) 2 different digits ...
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2answers
31 views

Number of ways to sit 6 girls and 6 boys together with no two girls together.

As the title of the question explains: What I thought on the very first instant was that we will make them sit alternate hence the answer will be 2 * 6! * 6! But ...
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1answer
20 views

four digit numbers that have at least one of their digits repeated

The number of four digit telephone numbers that have at least one of their digits repeated is A. 9000 B. 4464 C. 4000 D. 3986
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1answer
28 views

Finding the maximum possible order for an element in $S_5$

I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order. But my question is, do I have to write all elements of $S_5$, ...
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1answer
29 views

How many distinct elements does a group of permutation on 3 letters have?

I am having some problems solving a problem similar to this. So i tried making it a more simpler problem. I really don't know how to approach this kind of problem. A hint would be very much ...
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3answers
41 views

Forming 4-digit odd numbers under certain rules [closed]

How many four-digit odd numbers can be formed such that every $"3"$ in the number is followed by a $"6"$? A) 108 B) 2592 C) 2696 D) 2700
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37 views

finding the password [on hold]

Charlie has forgotten his six-digit ID number. he remembers the following: the first two digits are either 1,5 or 2,6, the number is even and 6 appears twice. if raju uses a trial and error process to ...
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0answers
29 views

why are these cosets equal?

Please disregard this question until I have uploaded a screenshot K is the subgroup of S_3 defined by the permutations {(1), (123), (132)} They have (1)K = (12)K = {(1), (12)} What they did was ...
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2answers
34 views

Coloring vertices of a square

Using four colors, red, white, blue and green, in how many ways can the vertices of a square be colored? Assume that reflections and rotations are allowed, meaning that if you examine a square from ...
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1answer
54 views

Number of seating arrangements in 5 cars

An exercise from Introductory Combinatorics by Richard A.Brualdi: A roller coaster has five cars, each containing four seats, two in front and two in back. There are 20 people ready for a ride. ...
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1answer
14 views

number of elements in unsortet case

I have a group M with Mn different elements. How many unique combinations can I make out of this in an n digit system when order is no importance. For example if M = {1 2} & n = 3 ...
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1answer
21 views

How many ways are there to place these books on the shelves?

You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.) I want to say $7^5$ since there are 7 possible shelves and ...
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Permutation help arranging letters

How many ways are there to arrange the letters A, B, C, D and E such that A never comes imme- diately after E or D and C always comes immediately before D? Help
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1answer
26 views

triangles and lines

There are 12 points in a plane. If 4 of them are on a straight line and no other 3 points are on a straight line, then find the difference between the number of triangles and the number of straight ...
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1answer
15 views

arrangement of balls in bowls

There are five bowls numbered $1$ to $5$. There are $5$ green balls and $6$ black balls. Each bowl is to be filled by either a green or black ball and no two adjacent bowls can be filled by green ...
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3answers
39 views

Writing a permutation group in 2 row notation

I have a permutation group in $S_7$, namely: $$(12345)(137)(56)$$ How do I write this in two row notation? I am to write it as disjoint cycles and then as transpositions but I feel better working in ...
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27 views

What is this probability that this event could occur?

Records show that 30% of the people, listed as passing the Firefighter's exam in Denver, are republican; the remainder as democrats. Last year, 30 people were hired as firefighters for the city. (25 ...
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0answers
19 views

What is total number of inversions in a permutation that is sorted except at indices that are a multiple of a certain number.

So say we have a permutation of integers that is indexed from 0 to n-1, and it is sorted in ascending order except for indices that are a multiple of a number call it x where x is smaller than all ...
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1answer
27 views

how can you count number of digits used in numbers from -2^127 to (2^(127) - 1)

There are numbers from -2^127 to (2^127)-1. I want to count the number of digits used in all the numbers. For example If I have numbers from -100 to 100 then number of digits used is $2*(1*9 + ...
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2answers
36 views

Kernel of $\phi:G \rightarrow \operatorname{Sym}(S)$ Group actions

$\operatorname{Sym}(S) == \text{All permutations of the set }S$. Prove $\ker(\phi)=\bigcap_{x\in S}G_x$ where $G_x$ is the stabilizer of $x$. Let $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
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1answer
129 views

How many possible combinations in 8 character password?

I need to calculate the possible combinations for 8 characters password. The password must contain at least one of the following: (lower case letters, upper case letters, digits, punctuations, special ...
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1answer
38 views

Number of permutations of AABBBCC, taking 7 letters at a time, when repititions are allowed

What is the number of permutations of the word AABBBCC, taking 7 letters at a time, repetitions being allowed? I think it should be $3^7$, but I can't see why. Also what would be the number of ...
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0answers
21 views

How many commutative block ciphers are there?

Let $K$ and $M$ and be two finite sets. Let $(G,\circ)$ be the group of permutations over $M$ under composition. Let a (implicitly: block) cipher with key in $K$ and message in $M$ be any application ...
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2answers
33 views

Difference between permutations

Given the following: 1) Is it wrong to say (1 2 4) (5 3) = (1 2 4) (5 3) or = (3 5) (1 2 3) ? 2) What is meant by ( 1 2 3 4 5 ) and 1 2 3 4 5 ? And why are they not equal? Thanks!
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1answer
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Permutation Multiplication (easy)

Given α◦β=(1532)(14)(35) How do we get from the given to = ( 1 4 5 2 ) ( 3 ) = ( 1 4 5 2 ) = (4 1 3 5 2) ? Thanks
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1answer
65 views

List all the permutations of {1,2,3,4}. Which are even, and which are odd?

The answer is: There are 24 permutations. The 12 even permutations are: id , (1 2 3 4) , (1 3 2 4) , (1 4 2 3) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3). The ...
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2answers
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Powers of permutation matrices.

Let $P$ be a permutation matrix obtained by the identity matrix by switching 2 rows $n$ times, (with no two rows switched more than one time). How to show that $$P^{\ n+1} = I$$? Is it true that, ...
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1answer
38 views

number of ways of constructing $n\times2$ rectangle from a $1\times2$ rectangle

You are given $1\times2$ rectangles and you have to construct an $n\times2$ rectangle from it. Tell the number of ways of constructing $n\times2$ rectangle from a $1\times2$ rectangle
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2answers
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A man, woman, boy, girl, cat, and dog are walking down a path..

I'm hoping someone can explain how this works. The problem: A man, woman, boy, girl, cat, and dog are walking down a path in single file. How many ways can this happen if the dog is between the man ...