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

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Permutation of composition factors?

Suppose $G$ is a finite length (or finite) group, with composition series $1 = G_0 \subseteq \ldots \subseteq G_k = G$. Denote the composition factors of this series (up to isomorphism) by the ordered ...
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2answers
66 views

Probability: What is wrong with my approaches?

$4$ out of $20$ balls are black. The others are white. If I pick $2$ balls out of those $20$ randomly, what is the probability that at least one of them is white? I could do this using a ...
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3answers
116 views

Dropping letters in post boxes

In how many different ways can 5 letters be dropped in 3 different post boxes if any number of letters can be dropped in all of the post boxes?
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2answers
78 views

Fill $8$ boxes with $60$ items

I have $8$ boxes and $60$ items: how many ways can I fill the boxes so that The order of the items in each box does not matter It does not matter which boxes are filled with which items. In other ...
3
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1answer
63 views

Irreducible representation of $\mathcal{S}_5$ over $\mathbb{C}$ of degree 4

I have come to a point where I need an irreducible representation of $\mathcal{S}_5$ over $\mathbb{C}$ of degree 4. Can somebody help me to find one and explain how to obtain one?
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1answer
31 views

Ordered Partitions

I'm having trouble figuring out exactly how one would get the answer for a question such as: "how many ways can 9 concert tickets be divided between 4 concertgoers, such that one person receives 3 ...
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3answers
392 views

Probability: deep understanding of permutations and combinations.

Look at this problem: If one is to throw $2$ dice, what is the probability that he/she gets a total of exactly $4$ dots? Well, the amount of the events that satisfy the condition is $3$ as we ...
2
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2answers
26 views

What's the name of the minimum number of transpositions required to build a permutation?

What's the name of the minimum number of transpositions required to build a permutation? I thought it was "rank" but apparently "rank" refers to the lexicographic number.
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1answer
45 views

Permutations Discrete Math

Roll a standard die n times. For finding the number of permutations of getting exactly k 3's you would do ${n\choose k} \cdot {5} ^{n-k} $ But what about finding at least k 3's? I was thinking of ...
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3answers
92 views

Conjugate to the Permutation

How many elements in $S_{12}$ are conjugate to the permutation $$\sigma=(6,2,4,8)(3,5,1)(10,11,7)(9,12)?$$ How many elements commute with $\sigma$ in $S_{12}$? I believe I use the equation ...
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3answers
537 views

How Many Permutations of Die Rolls Add Up to a Fixed Total?

I'm toying around with the idea of designing a board game. I like the idea of a hexagon-tile setup (a la Catan). There would be some number of different resource categories; I'm trying to determine ...
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1answer
56 views

Why do we swap the position in the cycle when writing disjoint cycles

I was watching a video on tranpositions and it isn't obvious to me why when decomposing a cycle, we swap the position of the elements in the cycle instead of swapping the elements themselves. I would ...
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2answers
31 views

Probability questions need help

1) You're going to a theme park with your family and they allow you to bring at least one but no more than 3 friends with you. If you choose from 8 friends, how many different grouping can be ...
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1answer
33 views

Number of ways of selection from a mixed group

For example I want to know number of ways I can select $2$ items from $AAABBC$ i,e $3$ of one group,$2$ of other and $1$ of last.
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1answer
44 views

All permutations from $S_6$ and $S_7$ by which $(1,2)(3,4,5)$ is conjugate to itself

The task is to find all permutation $\tau$ from $S_6$ and $S_7$ such that: $$\tau^{-1}(12)(345)\tau=(12)(345)$$ I think the answer is: $\{id \in S_6 , id \in S_7 , (67) \in S_7\}$ I would just like ...
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1answer
63 views

Basic combinatorics question, can you help me?

Before I start let me thank anyone that contributes to helping me with an answer for this. Ok - on to the question. Assume you have 10 jelly beans, 5 yellow and 5 red. What is the total amount of ...
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1answer
57 views

Forming a combination that is mathematically possible?

I have to implement an algorithm for a game. I will briefly explain the requirement for the team forming for the game. The game consist of two teams selected randomly from a pool of players. There ...
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1answer
20 views

Discrete Math Permutations help

Is it true that there are $5!$ possible sequences of five specific names? Wouldn't this be true since there are $5!$ arrangements?
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2answers
37 views

Find the no. of 4-digit no. that can be formed using $1,2,3,4,5$ if no digit is repeated .How many of these will be even?

I'm unable to solve this question please help. I have no idea of it and the answer given is 120 and 48 i got 120 but not 48 . this is how i got 120 no repeatation allowed so, ...
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1answer
54 views

Any two $n$-cycles are conjugate in $A_{n+2}$ if $n$ is odd

How would one go about proving the claim in the title? I see that if $\alpha,\beta$ are $n$-cycles and $\alpha,\beta$ permute $A,B\subset \{1,\dots, n+2\}$ respectively then $\overline{A\bigcap B}$ ...
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0answers
24 views

Security of permutation function

I have a question related to permutation. If I have a 10-bit number: {0101110010}. I want to swap even position bits to adjacent odd bits and odd positions to adjacent even bits, such that I get ...
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2answers
43 views

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$?

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$? The answer is given as $\{\mbox{id}, (1,2,3,4), (1,3).(2,4), (1,4,3,2)\}$. I understand how we got the first $2$ elements. Also ...
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1answer
18 views

Count the number of mixes of the card to get come back to origin

Mix a deck of 52 cards by placing them in two parts and take alternate cards from both stacks. Use the cycle notation to show that it only takes 8 mixes to come back to the origin. Since we want ...
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0answers
73 views

largest permutation group of odd order

For general $n$, what is the largest subgroup of the symmetric group $S_n$ that has odd order? I have a feeling that it might be the Sylow 3-subgroup...ADDED: but it isn't, as Mark Bennet points out ...
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1answer
42 views

Finding number of possibilities of n number of r combinations of which x can be unique behavior

I have say 5 alphabets altogether (a, b, c, d, e) out of which 3 (a,b,c) are from Bag 'A' and 2 (d,e) are from Bag 'B' I want all possible combination of 3 alphabets of these 5. so I used ...
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2answers
55 views

Permutations and probability related

A permutation of $1, 2, ..., n$ is chosen at random. Then the probability that the numbers 1 and 2 appear as neighbours equals (A)$\frac{1}{n}$(B)$\frac{2}{n}$(C)$\frac{1}{n-1}$(D)$\frac{1}{n-2}$ Is ...
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1answer
26 views

Prove that a representation have a base and it's irreductible

I'm quite new in representations and I'm trying to do next problem: (It's supposed that I don't know anything about characters theory) We want to study $S_3=(\tau=(123),\sigma=(1,2)\,|\, ...
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2answers
116 views

Calculating permutation of n items under pattern constraints

I am trying to find a formulae that gives the number of unique permutations of an arbitrary number of items, constrained with the following pattern: For every item X: X' must always appear at some ...
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5answers
305 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
2
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1answer
29 views

transitively action of stabilizer of G

if $G$ acts transitively on $X$ and for a special $x$ in set $X$ we have the stabilizer of $x$ acts transitively on $X-\{x\}$ can we conclude that this proposition is true for all element of $X$?
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1answer
95 views

Recurrence for the number of $\sigma \in S_n$ with cycle length at most $r$

I have just learned that the formula is right, but the definition of $c_n^{(r)}$ was wrong. The correct problem is: Prove $$c_{n+1}^{(r)} = \sum_{k=n-r+1}^n n^{\underline{n-k}} c_k^{(r)}$$ where ...
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1answer
97 views

How many homomorphisms are there from $\mathbb Z_3$ to $S_4$?

How many homomorphisms are there from $\mathbb Z_3$ to $S_4$? I got this on a quiz today and calculated 9 but I'm not certain I was right. I sent 1 to id and to the 8 3-cycles... did I do wrong?
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2answers
61 views

Number of ways to pick 3 people where no 2 of them are next to each other! [duplicate]

I've been trying to solve this problem, but i don't know what technique i should be applying and my answers are coming out to be wrong. I've tried putting them in different groups but non of them give ...
2
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1answer
309 views

Fixed points in random permutation [closed]

Suppose two random permutations of the numbers 1 to n placed side by side. a) Calculate the expectation number of fixed points for $n = 5$. b) Find the value of expectation in the amount of fixed ...
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0answers
23 views

Permutation and combinations count of members per group

A club with $x$ members is organized into four committees such that (1) each member is in exactly two committees, (2) any two committees have exactly one member in common. Then $x$ has (A) exactly ...
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1answer
61 views

Permutation cycles

My tasks are the following : Task 1 : Prove that $ \begin{pmatrix} 1 & 2 & \cdots & r-1 & r \end{pmatrix} = \begin{pmatrix} 2 & 3& \cdots & r & 1 \end{pmatrix} ...
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4answers
214 views

General approach for finding how many group homomorphisms are there

So I've asked this type of questions for more than once, and still I don't get the method(s) I've been presented with. What's the general recommended method for finding how many homomorphisms are ...
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2answers
36 views

A permutation problem (Kinda)

A collection of black and white balls are to be arranged on a straight line such that each ball has at least one neighbor of different color. If there are 100 black balls, then the maximum number of ...
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2answers
123 views

Find the number of ordered pairs $(m, n)$ of positive integers such that $mn = 2010020020010002$

No calculators allowed, so I don't know how to approach this problem, and I wonder if there's an easier way than factoring it all out by hand.
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1answer
53 views

2 questions about counting and permutation

I have a test on monday and I couldn't solve these 2 questions, I'd be grateful if you help me 1-) How many ways are there to distribute 18 balls among 6 di erent persons if a) each ball is di erent ...
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1answer
46 views

Prove the existence of a permutation for a matrix

Let $n\in\mathbb{N^+}, A $ be a matrix where $(a_{ij}) \in$ ${\{0,1\}}^{n\times n} $ so that the sum of each row or column of $A$ is $x$. For which $x \geq 1$ does a permutation $\sigma \in ...
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1answer
45 views

Modular arithmetic and maximal permutations

I have a research paper about pseudo-random number generators and I need to answer the following: Given $n \in \mathbb{N}$, let's consider the permutation group of $A=\{{0,1,\dots,n-1}\}$. Since $A$ ...
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1answer
46 views

Choosing a sample from a sample probability

I am a bit confused about this problem. I understand that you need to pick a sample first, K, and then find the probability of that sample being red, L. The total different combinations of picking a ...
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3answers
105 views

Group theory, conjugation of permutations

I have a past exam question that says... Decompose the following permutations into a product of disjoint cycles. Are the two permutations conjugate? $$\alpha= \begin{bmatrix} 1 & 2 ...
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4answers
551 views

In how many subsets of {1,2,3 … 9,10} there are odd number of objects from {1,2,3,4,5} and even number of objects from {6,7,8,9,10}?

In how many subsets of {1,2,3 ... 9,10} there are odd number/s of objects from {1,2,3,4,5} and even numbers of objects from {6,7,8,9,10} ? The answer I remember is 2^4 . 2^4 ( But It may not be ...
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2answers
243 views

In how many permutations of word “LOGARITHM” vowels have alphabetical order?

In how many permutations of word "LOGARITHM" vowels have alphabetical order? The answer is $9 \cdot 8 \cdot 7$ but I can't process it .
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1answer
53 views

Formalizing a permutation as a bijection between the same set?

The standard definition I usually find regarding permutations is a bijection from a set to itself, in other words: $\text{a function }f:\{ 1, 2,\dots,n \} \mapsto \{ 1, 2,\dots,n \} \text{ which is a ...
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3answers
142 views

Permutation & Combination Problem

I often solve math questions because I like it (This may sound crazy, I know :)). Today I came across an interesting permutation & combination question. The question is as follows: 6 people ...
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3answers
118 views

number of ways to sit 3 children in 5 available seats?

My math teacher put a problem on my review packet and she told us to cross it out because she couldn't figure it out herself. However, both of us knew it had a solution. The problem was as follows: ...
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0answers
92 views

finding the number of positive integral solutions

find the number of positive integral solutions of x+y+z=n, n belongs to set of natural numbers, n>=3 how far I've got: the number of positive integral solutions =coefficient of (x^n) in ...