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

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Minimum gems required to make a garland containing all permutations, with uniqe colored gems, of size n, where we have infinite gems of N colors.

What is the minimum number of gems required to make a garland (circular) which contains all permutations, with unique colored gems considered as a valid permutation, of size n, when we have infinite ...
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35 views

Identity element as product of transposition.

My lecture notes indicates that the identity element of a symmetric group $S_{n}$ is $\left ( \right )=\left ( 12 \right )\left ( 12 \right )$ Just to state: The identity element $b \in G$ ...
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1answer
23 views

Total possible combinations of variables

I have 4 factors, each containing a different number of entry. For example: I would like to list and compute the number of possible combinations. So each combination consist of 1 fruit, 1 drink, 1 ...
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1answer
25 views

RGB color combinations

RGB colors are selected by 3 selectors: Red, Green, and Blue. Each of these can be between $0$ and $255$. So (and I'm sure this is some kind of permutation but I can't put my finger on the actual ...
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1answer
25 views

Understanding proof for order of permutation

Theorem:The order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles. The proof for the order of permutation as extracted ...
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54 views

Permutation with constrained repetition: Distribution of random variable “number of pairs of identical elements”

You have a string of 360 letters: 180 x 'A' and 180 x 'B'. The number of ways this string can be permuted is $$\frac{360!}{180!180!} = \binom{360}{180}.$$ Assume the permutation is constricted ...
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There are $5$ women, $3$ men. How many ways to form a committee of $3$ with at least $1$ member of the opposite sex?

I have looked through several topics for similar solutions and I have attempted an answer to the question. Unfortunately, the sample question itself does not have an answer. From $5$ women and $3$ ...
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1answer
49 views

Permutations with limited repetition and various constraints

you have a string of 360 letters: 180 x 'A' and 180 x 'B'. I (hope I) understand that the number of ways this string can be permuted is $$\frac{360!}{180!180!} = \binom{360}{180}$$ What I have ...
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17 views

problem with alternating group with order 3

I was doing some computation with $A_3$, the alternating group on 3 letters. I know it has to be abelian, even cyclic, but when I carried out the actual computation... I couldn't make sense out of it. ...
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1answer
30 views

Product of disjoint cycles

I am working on a problem and it ask to solve for the product of disjoint cycle from left to right convention. Have I done anything wrong in my attempt? $$(16527348)\cdot (152468)\cdot( 37 )$$ ...
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22 views

Permutation of vertices of a square $(D_{4})$

$D_{4}$, the dihedral group of order 8 is a square. Suppose the vertices are labelled in an anti-clockwise fashion, starting with 1 at the bottom right corner. Each rotation is $\frac{2\pi}{4}$=90 ...
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4answers
77 views

selecting 4 non-consecutive books from 10 books.

I have a set a $10$ book kept in a line and I want to find out how many ways $4$ books can be chosen from that if I don't choose consecutive books to be taken out. I felt this is similar to ...
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1answer
15 views

Rolling Dice Probability

A fair dice is rolled 3 times, The probability of the product of the three outcomes is a prime number is? The products which give a prime number I found out to be only 4. However for the total ...
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2answers
98 views

The set of all even permutations in G forms a subgroup of G

Show that if $G$ is any group of permutations, then the set of all even permutations in $G$ forms a subgroup of $G$. I know that I need to show the closure, identity, and inverses properties ...
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1answer
17 views

circular permutation in case of identical diamond and pearls

the number of ways a necklace can be formed by 18 identical diamond and 3 identical pearl ? my solution is that divide the procedure into three case case1:all pearls together that leads to only one ...
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1answer
38 views

Seating Problem ( Permutations and Combinations )

What is the number of ways seating three gentleman and three ladies in a row such that each gentleman is adjacent to one lady ? My attempt - My attempt was to first make three groups each containing ...
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1answer
88 views

Permutation with no adjacent elements.

There are 2 Red balls, 3 Blue balls, 4 Green balls, 2 Yellow balls. In how many ways can we place the balls in straight line such that no two identical balls (balls of the same color) are in adjacent ...
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1answer
60 views

What is the probability that a five-card poker hand has four ACES?

What is the probability that a five-card poker hand has four ACES? When I was solving the above stated problem, I got confused while trying different methods : Assume a normal $52$ deck of ...
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516 views

How many ways are there to make a schedule of 6 subjects

There are 6 different subjects including maths and physics. How many ways are there to make a schedule of 6 subjects such that physics follows maths? Actually I tried to denote maths as M and ...
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0answers
42 views

Parity of a Permutation Significance

I have a permutation of N numbers.I can perform these operation any number of times in order to sort the permutation. Choose any 3 consecutive indices and rotate ...
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1answer
26 views

What does each element of the Klein 4-group represent?

I'm new to the concept of the Klein 4-group. I am familiar with the concept of the alternating group and what elements like (1 2 3) represent. However the Klein 4-group contains elements like (1 2)(3 ...
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1answer
41 views

Arranging 12 people in a circle-like table [closed]

Based on the picture, I want to solve Problem ⓐ. Is the problem ⓐ same problem compared to problem ⓑ? So we can suppose one of the people is a King, then the answer should be 11! Am I right? :o
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48 views

Finding no. of cases where $12$ people finished the Marathon

$12$ people ($6$ male, $6$ female) run a marathon. Every time a person finishes a marathon, males who finished marathon are more than or equals to females who finished it. How many number of cases ...
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136 views

How do Gap generate the elements in permutation groups?

I understand that permutationgroups in Gap are represented by generators, which seems to be far more efficient than groups represented by all it's elements, but how could then for example ...
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1answer
22 views

Finding the order of permutation

I am making a mistake in solving the following problem. I would really appreciate it if someone looked over it and helped me with the solution $a=(1,2,3)(2,3,4)(5,6,7)(7,8,9,10)$ I need to show that ...
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2answers
40 views

What are all the elements of the group of symmetries of a regular tetrahedron?

I can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on 'top' and we do to the other $3$ as we did in a ...
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2answers
29 views

How many ways are there to choose 2 numbers such that their product is a multiple of 3

Let A be the set A = {1; 2; 3; ... ; 20} containing natural numbers from 1 to 20. How many ways are there to choose 2 numbers for A such that their product is a multiple of 3? I tried to take the ...
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1answer
16 views

size of group of row and column flips of a square board

Let $X$ be the set of numberings of the squares in a $n \times n$ board with the numbers $1$ to $n^2$. Let $G$ be the group of transformations of boards generated by row and column flips, where a flip ...
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1answer
42 views

How many combinations of permutation matrices are there?

I want to know, for an $n\times n$ permutation matrix, how many matrices are there such that there are exactly $3$ entries above the diagonal. For example, there is only one $4\times 4$ matrix that ...
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1answer
30 views

Arranging m distinct groups of k elements each?

There are n elements in total which is already divided into m groups of k elements each. Thus, n=m*k. The question is, how many arrangements of these m groups are possible? I came to a possible ...
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1answer
39 views

Combinatorics - Selecting Finite Possible Pairs from Infinite Pool

A friend of mine asked me this puzzle few days back. I have been trying to solve this. I tried it by doing manually creating possible pairs of yellow & blue ribbons But I am sure there has to be a ...
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1answer
43 views

find normal subgroup of symmetric group

Symmetric group $S_3=\{(),(1,2),(2,3),(1,3),(1,2,3),(1,3,2)\}$, I understand that $H=\{e,(1,2,3),(1,3,2)\}$ is the normal subgroup of S3 ($H\lhd S_3$) because: $$ gH=Hg, \forall g\in S_3$$ e.g. let ...
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Prove that every permutation in $S_k$ is the product of transpositions of the form $(j, j + 1).$

Prove that every permutation in $S_k$ is the product of transpositions of the form $(j, j + 1).$ I proved the case $n=2$ for my base case... so $(12)=(21)$ and $(21)=(12)(12)$ then I proved $n=3$ and ...
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3answers
42 views

The number of words containing four $a$'s and two $b$'s

Find the number of words containing four $a$'s and two $b$'s. I thought of $6!$ but then I found out that there will be many repetitions in that case. Thanks in advance.
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1answer
22 views

Parallel lines with points [closed]

There are 2 parallel lines. One of them has 5 points on itself, and the other one has 4 points on itself. How many triangles are there whose vertices are those points. Thanks in advance.
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How many ways are there to choose $6$ children from $7$ boys and $4$ girls on condition that at least one is a girl? [closed]

There are 7 boys and 4 girls in the kindergarten. How many ways are there to choose 6 of them on condition that at least one is a girl. Thanks in advance.
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1answer
21 views

How to find all stabilizater of element of the same orbit.

Let $\mathfrak S_4\times \mathfrak S_4\longrightarrow \mathfrak S_4$ the action by conjugaison, i.e. $$\sigma \cdot \tau=\sigma \tau\sigma ^{-1}.$$ I have shown that the orbits are $\mathcal ...
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1answer
27 views

Permutation and induction

Each permutation in $A_k$ can be written as a product of 3-cycles of the form (1, 2, 3), (1, 2, 4),...,(1, 2, k). I am trying to start this problem by induction but I am having trouble with the base ...
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1answer
29 views

Writing permutation as a product of transpositions

I have a problem writing permutations as a product of disjoint cycles. For example, in the book, there are the following cycles: $(132)=(13)(12)$, $(1243)(243)=(23)(34)(14)$ Can someone please ...
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28 views

Counting powers of permutations

I didn't find similar questions so decided to ask this one. Given positive integers $n$ and $d$ how can we efficiently estimate (or better calculate) cardinality of the set $~~ \{ \sigma^d ~~|~~ ...
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1answer
31 views

Combinations of letters with restrictions

Create a string of five letters using the letters: A, B, C, D, E, F, G, H, I, J, K, L, M. a) How many words contain at least one ...
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Poker Probability. Multiple Questions.

this is for my own personal use, not school related. Texas Holdem. Each player is dealt 2 cards from a deck of 52 cards. Once the cards are shuffled the dealer gives each of the 9 players 2 cards, ...
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1answer
35 views

Interesting sequence of all the natural numbers [closed]

What are some sequences that contain all of the natural numbers that come up naturally in mathematics? (Obviously, there are an infinite number of sequences of all the natural numbers ($2^{\aleph_0}$ ...
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0answers
12 views

Verify $T^*(f^\sigma)$ = $(T^*f)^\sigma$

Where $T^*$ is linear. $f^\sigma(v_1,...,v_k)$ = $f(v_{\sigma(1)},...,v_{\sigma(k)}) $T^*f(v_1,...,v_k)$ = $f(T(v_1),...,T(v_k))$ Attempt at the proof: I didn't use the fact that T is linear ...
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75 views

arrangement of $n$ oranges and $n$ apples around a circle

what is the total number of distinct arrangements of $n$ oranges and $n$ apples around a round table? I have no idea how to go about.
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1answer
47 views

how many words can be formed from a scrabble rack with 7 letters

Given a scrabble rack with 7 unique letters, how many words (meaning not important) can be formed with 1 to 7 letters? My first thought was to take all the permutations from p(1,7) to p(7,7) and add ...
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1answer
29 views

string and its permutations

I have a string lets say abcd so its all permutations would be ...
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15 views

Write $\sigma$ as a composite of elementary permutations

Let $\sigma \in S_5 $ be the permutation s.t: $(\sigma(1),\sigma(2),\sigma(3),\sigma(4),\sigma(5))$ = (3,1,4,5,2) Write $\sigma$ as a composite of elementary permutations. Definition of elementary ...
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21 views

How do I show that the two methods of permutation test are both the same?

My main objective is to show the methods described below are really the same. However, I am having difficult both formulating the idea clearly and proving my assertion. Below is my attempt. Suppose ...
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5answers
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Number of ways of making a die using the digits $1,2,3,4,5,6$

Find the number of ways of making a die using the digits $1,2,3,4,5,6$. I know that $6!$ is not the correct answer because some arrangements can be obtained just by rotation of the dice. So there ...