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

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1answer
18 views

Finding permutation $a$ given $b$ and conjugate $a^b$

Normally we define a conjugate relationship as $$a^b = b~a~b^{-1}$$ But I don't know how to find $a$ given that we know $b$ and $a^b$.
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1answer
19 views

What is the minimal cardinality for a generating set of the permutations?

I want to find the minimum number of permutations so that all other permutations can be obtained by multiplying the permutations of this set (taken in any quantity). In other words, I am looking for ...
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1answer
10 views

Number of subsequences in a string

I know this might be one of the silliest questions out there but I'm going ahead and ask it here since I've lost practice in mathematics. I have been reading that the number of subsequences in a ...
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0answers
12 views

conjugated permutations as to solve

We have: Niech $\sigma = (13624)(587)(9), \tau=(15862)(394)(7) $ Determine such permutation $\alpha$ that $ \alpha \sigma \alpha^{-1} = \tau$ How much are they?
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1answer
23 views

Order of a permutation divides n in Sn

Let $\theta \in S_n$, and for any $k \in \mathbb{N}$, either $\theta^k = I_{I(n)}$ or $\theta^k$ has no fixed elements. Show that $o(\theta) | n$. $I_{I(n)}$ denotes the identity. I'm completely ...
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1answer
25 views

Question in permutations

When we use this law? And in any case we use it? Thank you and I wish clarification.
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1answer
22 views

Permutations and their cycles [on hold]

What is the probability that k given elements belong to the same cycle in random permutation ? Thank you in advance.
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0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
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2answers
31 views

Probability, why is this wrong? (Combinations and Permuations)

Why is this the wrong approach to solve this problem? "There are 65 students. 20 of them are sophomores, 20 are freshmen, 15 are juniors and 10 are seniors. When picking a 4 student committee, ...
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1answer
23 views

N Boxes and M babies question.

There are N boxes placed in a straight line. Adjacent boxes are separated by 1 unit. The Babies which are a total of M in number decide to play in this arena of boxes by moving from one box to ...
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2answers
18 views

In how many ways can a coat be chosen such that exactly one person picks up the correct coat?

There are n people at a party. At the end they each take a coat at random. a) How many ways can coats be chosen such that no person picks up their own coat/what is the probability that no person ...
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1answer
31 views

Find the number of elements of order 3 in $S_7$

I understand that there are two cycles of length 3, $(i,j,k)(a,b,c) \in$ $S_7$. However, I'm quite stumped in figuring out the logic behind these steps, leading to the answer : Number of distinct 3 ...
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1answer
18 views

In $S_6$, write the result as a product of disjoint cycles and then in the 2-row form.

(a) $(1,2,4)(4,3,5)(2,4)(1,2,4)^{-1}$ In the solution for this question, my professor has the product of disjoint cycles written as (1,3,5)(4,1). How would this make sense when disjoint cycles are ...
0
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1answer
15 views

permutations combinations

Q1. Total number of permutations of k diferent things , in a row , taken not more than r at a time(each thing may be repeated any no. of times) is equal to Q2. A teacher takes 3 children from her ...
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1answer
26 views

Show $\alpha^m = \varepsilon$ working with permutation groups

Show that $\alpha^m = \varepsilon$ using $\alpha^\ell (a_i) = a_{(i+\ell) \bmod{m}}$ where $\alpha = (a_0 a_1 \dots a_{m-1}) \in S_n$ a permutation group. I've been working on this problem but can't ...
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1answer
24 views

How to generate a single instance of multichoose (stars and bars)

So we know that if I have $k$ balls and $n$ buckets, I have $\binom{n+k-1}{k}$ unique ways to allocate the balls. Let's say $n=4$ and $k=2$ then I have $\binom{5}{2}=10$ ways. All possible allocations ...
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1answer
115 views

Pixel Permutations

How many possible arrangements of pixels can a 1024x768 pixel screen display if the color of a pixel is determined by mixing 3 values: red, green, and blue, ranging from an intensity of 0 to 255? The ...
2
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1answer
71 views

Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same?

Given n1 number of 1's, n2 number of 2's, n3 number of 3's, n4 number of 4's. form a sequence using all these numbers such that two adjacent numbers should not be same. I have tries lot of things ...
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0answers
11 views

Permutation of conjoined faces in regular polygon with diagonals

I've been doing some study on relationships in polygons, right now, regular polygons. I've been trying to find relationships between the diagonals, angles, faces, vertices, and primarily conjoined ...
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3answers
71 views

Permutations and Combinations? 3 digit number…

This is a question of permutation and combination. Q. How many words can be formed from the word "LUCKNOW" when i) No restriction is there ii) L is the first letter of the word iii) All the vowels ...
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1answer
59 views

Permutations and Combinations. Arranging things to be adjacent etc…

Given an empty stack S and a range Rn of integers [1,n]. Consider a sequence S' of n push operations and n pop operations on S in any arbitrary order such that push(i) always precedes push(i+1) and ...
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0answers
15 views

Number of solutions for a multiple traveling salesman (mTSP) problem

Traveling salesman problem (TSP) with n-number of cities and only one salesman has "nPn" solutions which is n! but when you have more than one salesman, say k-number salesman, to travel n-number of ...
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1answer
13 views

Determine the isometric group $G$ which transfers a square into it self

I am solving the following exercise: Determine the isometric group $G$ of the euclidean plane which transfers a square into it self. The restriction of an element $g \in G$ on the vertices of ...
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3answers
26 views

Computing the inverse of a permutation

I didn't understand the permutation and of course, I got this question wrong. Compute the inverse of the following permutation: $$ \begin{pmatrix} 1&2&3&4&5&6\\ ...
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2answers
86 views

Permutaion and Combination Problems

Hey folks I'm having some issues with permutation and combination problems. 1) We have: Niech σ=(13624)(587)(9),τ=(15862)(394)(7) Determine such permutation α that ασα−1=τ How much are they? I ...
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1answer
20 views

$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ contains a $5$-cycle?

Context: Consider $$ f(x):=x^5-4x+2\in\mathbb{Q}[x]. $$ By Eisenstein's criterion, $f$ is irreducible over $\mathbb{Q}$. Since $\mathbb{Q}$ has characteristic $0$, we know every irreducible polynomial ...
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2answers
43 views

Number of ways to group digits in {1,2,3,4,5,6,7,8,9} into numbers, while maintaining order

I have a set of integers from 1 to 9, call it A: $$A=[1,2,3,4,5,6,7,8,9]$$ How could I find the total number of possible combination of numbers within that set, while maintaining order? For example, ...
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1answer
25 views

Letter Arrangement with Permutations

In how many different ways can the letters of the word MAMMAL be rearranged so that the letters M are separated?
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1answer
32 views

How to find out if two cycles are conjugate?

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$. I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for ...
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3answers
37 views

$V_4\triangleleft S_4$

Let $V_4:=\{(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3),\iota\} \leq S_4$. It is possible to show $V_4\triangleleft S_4$ by considering conjugation. However, after long thought on the matter, I don't ...
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1answer
21 views

Painting parallelopiped with 6 different color

In how many ways can 6 faces of a rectangular parallelopiped with all 3 dimensions distinct , be painted with 6 different colours?? I have tried and i am getting 90 by $\displaystyle\frac{6!}{2^3}$. ...
2
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3answers
53 views

How many ways can the average of n dice be a?

For example, if n = 2, and a = 3.5 one could have (1,6), (2,5), (3,3), (4,3), (5, 2), (6,1) = 6 if n = 10 and a =3.5, one possible combination could be (1, 6, 1, 6, 1, 6, 1, 6, 1, 6)
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1answer
31 views

Partial Derangements

There are n people and n houses, such that every person owns exactly one distinct house. Out of these n people, k people are special (k<=n). You have to send every person to exactly one house such ...
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2answers
59 views

show elements of order 15 in $S_8$ satisfy a relation

Let $G$ be a group. Define a relation $\sim$ on $G$ by $a \sim b$ if there exists $g \in G$ such that $a = gbg^{-1}$. Prove that all elements of order 15 in $S_8$ are related by $\sim$. I noticed ...
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1answer
37 views

solving a simple problem of combination , with different approach

I found a question and I have different approach to solve it , but unable to get the answer. Question :How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible ...
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2answers
53 views

How many permutations of N integers with K pairs of numbers that can't be adjacent?

This is a computer science problem, I have a difficulty with the math part. There are $n$ integers $\{1, 2,\dots, n\}$ and $K$ pairs of numbers $(a, b)$; $a \ne b$; $a, b \le n$. No pairs are ...
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0answers
19 views

arranging three objects in six spaces

How many ways are there to arrange three objects in six spaces so no two objects are next to each other? I know the answer is 4 by doing it manually How can you tackle this problem using ...
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1answer
58 views

probability n! please help it's the only question i don't know how to do on the homework [closed]

write n! in terms of (n-1)! I am not sure what it is asking. I have tried everythng.
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1answer
16 views

Permutation of word

Question: Find the permutation of letters of the word EXERCISES in which vowels are together. My Efforts: I have rearranged the word in such a way that all the vowel come together. EEEI XRCSS Now ...
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3answers
27 views

Probability with colored flags

A signal has $6$ flags, each flag can be blue, white or red. Possible signals formed is $n^r = 3^6 = 729$ possible signals formed How many different signal can be made from $6$ flags of which $3$ ...
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3answers
48 views

round table seating probability

There are $6$ people, let's call them - (a,b,c,d,e,f), to sit at a round table. The number of ways they can arrange themselves is $(6-1)! = 5! = 120$ ways. What is the probability that person 'a' ...
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0answers
23 views

Permutation combination question

Say that i have 4 variables: $X_1, X_2, X_3$ and $X_4$. and $Y$ is a function of these four variables which uses four operational symbols $(+,-,*,/)$. This could be: $X_1+X_2+X_3+X_4 $ ...
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2answers
37 views

Expressing permutations as products of transpositions and identifying parity

$(156)(234)= (16)(15)(24)(23)$ Even $(17254)(1423)(154632)= (12)(13)(16)(14)(15)(13)(12)(14)(14)(15)(12)(17)$ Even How is parity determined? Is it simply even when the # of transpositions is even ...
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2answers
24 views

If $n>m$, then the number of $m$-cycles in $S_n$ is given by $\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}$.

Show that if $n>m$, then the number of $m$-cycles in $S_n$ is given by $$\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}.$$ My doubt Suppose I wish to count the number of $m-$cycles. Then I will get ...
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1answer
24 views

Set combinations

Assume all strings of length 3 from the set S = {A,C,G,T}, (allowing repetition) are equally likely. What is the probability that such a string has no C given that it has no A? Totally lost with ...
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1answer
143 views

permutation and f(n) challenge

Suppose $f(n)$ be the number of permutation from set ${1,2,..,n}$ such that for each $ 1 \leq i \leq n$ we have: $ | \pi(i)-i| \leq 1 $. meaning of $ \pi(i)$ is an elements whose in place $i$ of ...
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3answers
40 views

Subgroups of $S_4$ generated by cycles

I am new with abstract algebra and I trying to find all the subgroups of $S_{4}$ generated by the cycles : a) $(13)$ and $(1234)$ b) all cycles of length $3$ I am not sure how to start so I would ...
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3answers
107 views

A question in permutation

Help is needed in solving the following problem. $8$ persons ($A$ and $B$ and $P, Q, R, S, T, U$) are to be seated in $2$ rows ($4$ seats per row). Find the number of ways that $A$ and $B$ are ...
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1answer
36 views

Equation for a systematic permutation

A $6$ digit number is set whereby every digit can be repeated without any constraints. So one can have a number between $000001$ and $999999$. (Zeros on the left are counted). The problem: Generate ...