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

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0
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3answers
35 views

Number of shoes to be taken out.

There are $7$ pairs of black shoes and $5$ pairs of white shoes. They are all put into a box and shoes are drawn one at a time. To ensure that at least one pair of black shoes are taken out, ...
2
votes
1answer
40 views

Combination of N-number list with M enabled items

I have a list of N numbers (boolean), where a exactly M of them must be selected (true), not more or less. An item of the list can only be selected (1) or not selected (0). Example with N=4, M=3: [...
0
votes
0answers
44 views

Find all permutations of length n from a set of m values

If I have a set of values, say, [0, 1, 2, 3, 4], how can I find all permutations of a given length (that is less than the number of values in the set)? For example, how would I go about finding all ...
3
votes
1answer
35 views

Number of orderings with tree-based constraints

How can I compute the number of possible orderings of the numbers $1,\dots,N$, where some constraints are given on the relative order of some numbers? I know how to do the calculation in simple cases....
3
votes
1answer
25 views

The wreath product $K ~\mbox{wr}_{\Gamma} ~ H$ acts faithful on $\Delta \times \Gamma$ iff $K$ acts faithful on $\Delta$

Let $K$ and $H$ be groups, and let $H$ act on $\Gamma$. Also let $\operatorname{Fun}(\Gamma, K) = \{ f : f : \Gamma \to K \}$ be the set of all functions from $\Gamma$ to $K$. This set is a group ...
0
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2answers
181 views

6-team Round Robin

Is it possible to create a schedule where we have 6 teams play 5 different games with each team playing each game and each team only once? Or do we have to increase the amount of games to make this ...
1
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1answer
39 views

find coordinates are of a parallelogram

if the coordinates are (1, 2) (4, 3) (1, 0) (-2, -1), how can we can find whether it is of a parallelogram or not
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2answers
46 views

If $y$ is a cycle of length $r$, show that $\sigma y \sigma^{-1}$ is also a cycle of length $r$.

I am trying to show that if $y$ is a cycle of length $r$, and $\sigma \in S_n$ then $\sigma y \sigma^{-1}$ is also a cycle of length $r$. More specifically, that if $y = (k_1\ \dots k_r)$ then $\sigma ...
3
votes
3answers
45 views

Number of Permutations of 30 numbered balls with restrictions

In how many ways can one arrange 30 numbered balls in a row, so that: balls 1-10 are not in places 1-10 balls 11-20 are not in places 11-20 balls 21-30 are not in places 21-30 I tried ...
0
votes
4answers
69 views

Number of positive numbers of not more than $ 10$ digits formed by $0,1,2$ and $3$.

The number of positive numbers of not more than $10$ digits formed by $0,1,2$ and $3$ is. $\color{green}{a.)\ 4^{10}-1} \\ b.)\ 4^{10} \\ c.)\ 4^{9}-1 \\ d.)\ \text{none of these} $ I tried , ...
1
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1answer
11 views

Showing that the $T_n + T_{n+1}$ equals the number of ways of occupying something different

I am having trouble with the wording of this question. $Tn$ is the number of derangements of a set of size $n$. $n$ treeloppers are cutting down $n+1$ trees. They go and have a lunch break in the ...
2
votes
4answers
624 views

probability that exactly two envelopes will contain a card with a matching color

Suppose that 10 cards, of which 5 are red and 5 are green, are placed at random in 10 envelopes, of which 5 are red and 5 are green. Determine the probability that exactly two envelopes will contain a ...
6
votes
1answer
83 views

With how many ways can we place 20 cars in 30 spots.

So we have 4 white cars, 6 black cars, 6 blue cars and 4 silver cars. We want to place them in a 30 spot parking. We choose to place the cars with the following order white cars first, then silver ...
0
votes
1answer
171 views

Permutations with 26 letters

Not totally sure if I'm understanding the questions correctly: Consider the permutations of the set of 26 letters of the English alphabet How many total permutations are possible? P(26,26)=26! How ...
1
vote
1answer
358 views

Finding value of unknown in factorial equation

What's the value of $n$ in the following equation? $$2(2n-4)!= (n-4)!(n+2)!$$ I've tried coming up with an equivalent combination and expanding $(n+2)!$ but that didn't get me anywhere.
2
votes
2answers
46 views

On the Commutativity of Cycles in Permutation Groups

Let $\sigma,\tau$ be two cycles in $S_n$ such that $\sigma$ and $\tau$ have different length; $\sigma$ and $\tau$ are not disjoint. Then is it always true that $\sigma\circ \tau\neq \tau\circ\...
1
vote
2answers
186 views

16 teams enter a competition

16 teams enter a competition. They are divided into four pools (A, B, C and D) of four teams each. Every team plays one match against the other teams in its pool. After the pool matches are completed:...
1
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2answers
296 views

In how many ways can a group of 5 boys and 5 girls be seated in a row of 10 seats?

In how many ways can a group of 5 boys and 5 girls be seated in a row of 10 seats? Still having some confusion with the difference between combinations and permutations. I have tried this problem ...
1
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0answers
21 views

How can we derive Derangement Theorem? [duplicate]

can anyone please help me to derive the given result for finding number of ways in which N objects can be arranged such that no object goes to their proper place which is given by $N!\left(1-\frac{1}{...
0
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0answers
39 views

What's the difference between product of disjoint cycles vs cycles for permutations?

I'm getting very confused by products of cycles vs disjoint cycles. They look exactly the same to me and the way to compute them seems the same.
-3
votes
3answers
385 views

The word “SQUARE” [closed]

Assuming that any arrangement of letters forms a 'word', how many 'words' of any length can be formed from the letters of the word "SQUARE"? No repeating of letters.
0
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0answers
50 views

Number of cycles in shift permutation

Suppose we're given the following permutation of $n$ first positive integers: $n-k+1, ..., n, 1, ... k$ where $k \leq n$ Experimenting with different $n$ and $k$ i found out that it has $gcd(n,k)$ ...
1
vote
1answer
116 views

How many sequences possible in stack , if the input(1,2,3,…,n) is in order?

How many permutations can be obtained in the output (in the same order) using a stack assuming that the input is the sequence $(1, 2, 3, 4, 5,\dots, n)$ in that order? Example If $n=5$, then ...
9
votes
5answers
243 views

In how many ways can a selection be done of $5$ letters?

In how many ways can a selection be done of $5$ letters out of $5 A's, 4B's, 3C's, 2D's $ and $1 E$. $ a) 60 \\ b) 75 \\ \color{green}{c) 71} \\ d.) \text{none of these} $ Number of ways ...
0
votes
0answers
49 views

Show that if $\sigma \in S_n$, $\sigma^2 = \epsilon$ iff $\sigma$ is product of disjoint transpositions.

I am trying to show that if $\sigma \in S_n$, $\sigma^2 = \epsilon$ iff $\sigma$ is product of disjoint transpositions. This is my attempt: Suppose $\sigma = (k_1\ k_2)\dots(k_{r-1}\ k_r)$ then the ...
6
votes
1answer
55 views

In how many ways can '$6$' things be distributed equally among $2$ groups?

In how many ways can '$6$' things be distributed equally among $2$ groups ? I tried $\dbinom{6}{3}\times \dbinom{3}{3}\times 3!\times 3!$ But I am not sure if it is correct . I look for a ...
1
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3answers
63 views

Arranging $100$ balls.

There are $100$ balls numbered $n_{1}, n_{2}, n_{3}\cdots \cdots n_{100}$ . They are arranged in all possible ways . How many arrangements would be there in which $n_{28}$ ball will always be ...
2
votes
1answer
38 views

Prove that $A_{n+1} \cap S_n = A_n$ with permutations.

I am trying to prove that $A_{n+1} \cap S_n = A_n$ for $n>2$ and n is an integer. Here $A_n$ is the Alternating Group of degree $n$ and $S_n$ is the symmetric group. It seems as if induction ...
2
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1answer
37 views

Problem with the proof of a theorem in the field of permutation groups

My question is about the proof of the Theorem $3.4B$ of the book "Permutation groups" by J. Dixon and B. Mortimer. I understand the proof with assumption (i) but my problem is with assumption (ii), ...
0
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0answers
162 views

A problem involving combinations and permutations

How many committees of 4 can be formed from 17 republicans and 11 democrats if 2 members of each party must be on the committee? I slightly know how I have to go about to solve this but I can't keep ...
2
votes
2answers
92 views

Forming a committee from $4$ gentlemen and $4$ ladies with certain conditions

From $4$ gentlemen and $4$ ladies a committee of $5$ is to be formed . If the committee consists of $1$ president, $1$ vice president and $3$ secretaries. What will be the number of ways of ...
0
votes
1answer
61 views

Forming a committee from $4$ gentlemen and $4$ ladies with conditions.

From $4$ gentlemen and $4$ ladies a committee of $5$ is to be formed . Find the no of ways of doing so if the committee consists of $1$ president, $1$ vice president and $3$ secretaries? $a.)\ ...
1
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1answer
70 views

Better explanation of conjugation of permutations?

It says here that $\rho = \tau \sigma \tau^{-1}$, and if $\sigma(i) = j$, then $\rho(\tau(i)) = \tau(j)$. So take for example $\sigma = (1 3 2 4) (5 6)$ and $\rho = (5 2 3 1)(6 4)$. To find $\tau$, it ...
1
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2answers
1k views

A restaurant offers 5 choices of appetizer, 10 choices of main meal and 4 choices of dessert.

A customer can choose to eat just one course, or two different courses, or all three courses. Assuming all choices are available, how many different possible meals does the restaurant offer? So here ...
0
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0answers
48 views

Permutations of sets within larger sets

I am trying to find a formula to give me the number of permutations of 3 numbers across multiple sets of numbers. For example I have 5 different sets call them a,b,c,d,e with numbers 1-10 and I want ...
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votes
1answer
45 views

$4$ digit numbers divisible by $5$ formed with $0,1,2,3,4,5,6,6$

How many $4$ digit number divisible by $5$ can be formed with the digits $0,1,2,3,4,5,6$ and $6$? $a.)\ 220\\ \color{green}{b.)\ 249}\\ c.)\ 432\\ d.)\ 288 $ I tried Case $1$. $\text{ _ _ _ 0}$...
0
votes
1answer
28 views

Permutation with repetition and condition

Given an integer $N$, Find how many strings of length $N$ are possible, consisting only of characters A, B, C with each character occuring at least once. Solution: Lets place A,B,C (at least once) ...
2
votes
2answers
118 views

The total number of games played in chess tournament

There are two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to ...
0
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1answer
78 views

Squares in chessboard that must not lie in the same row or same column

In how many ways is it possible to choose a white square and black square on a chessboard so that the squares must not lie in the same row or same column? $a.)\ 56 \\ b.)\ 896 \\ c.)\ 60 \\ \...
2
votes
1answer
35 views

Compute $w^2 $ if $w = 782193456$ in cycle notation and express the answer in terms of generators.

Compute $w^2 $ if $w = 782193456$, in cycle notation and compute $w^2$ using its expression in terms of generators $s_1...s_8$, simplifying it as much as possible. my answer is $w^2 = (147)(256)(389)...
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3answers
51 views

A problem about the alternating group $A_4$

How can I prove that $A_4=\{\sigma^2:\sigma\in S_4\}$? My approach is the following: If $T:S_4\to S_4$ is defined by $T(\sigma)=\sigma^2$, then we have to prove that $T(S_4)=A_4$. Since $sgn(\sigma^2)...
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votes
1answer
32 views

122233 how many permutations?

I have a simple question: How many six-digit numbers have the same digits as the number 122233? I'm stuck on a such a simple problem.
1
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0answers
41 views

Graph isomorphism in terms of permutation matrix elements

The graph isomorphism problem is defined as follows. If $\Gamma_1$ and $\Gamma_2$ are two graphs with adjacency matrices $A_1$ and $A_2$ respectively, is there a permutation $\pi$ such that $...
1
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1answer
47 views

Is my assumption about dependencies for this particular setup correct?

Given a list of positive, whole numbers $n_1, n_2, \cdots, n_q$ and $m\in\mathbb{N}$, such that $0 \le n_k < m$ for $k = 1, 2, \cdots, q$, let $T$ be the set of all possible tuples $\left(t_1, t_2, ...
1
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3answers
51 views

If $G$ acts on $\Omega$ and $|G| = |\Omega| + 1$, show there exists nontrivial element fixing point without Burnside's lemma

Let $G$ act on $\Omega$ transitively, and let $|G| = |\Omega| + 1$ (both sets are assumed to be finite). I want to show from first principles (using maybe arguments like the pigeonhole principle, but ...
0
votes
0answers
40 views

Bijection between tensors and permutations (in linear $O(n)$ time)

The number of permutations of the set $S=\{1, \dots, n\}$ is $n!$, or in other words the permutation group $S_n$ has $n!$ elements The number of tensor components of a tensor in $n$ dimensions $(d_1=...
1
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1answer
35 views

A primitive permutation group with abelian point stabilizars

Suppose that $G$ is a group acting primitively on a set $X$. Also suppose that all of its point stabilizers are abelian. I want to prove that $G$ is either regular of prime degree or a Frobenius ...
0
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3answers
101 views

Finding number of roads

There are three cities $a,b,c$. Each of these cities is connected with the other two cities by at least one direct road. If a traveler wants to go from one city to another city, she can do so ...
1
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2answers
101 views

How many ways are there to split $n$ students into groups of size $x$ OR $y$ ($50$ students into groups of $5$ or $6$)?

How many ways are there to split $n$ students into groups of size $x$ OR $y$ ($50$ students into groups of $5$ or $6$)? I understand there might not be an equation in general but is there an ...
1
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2answers
52 views

Formula for permutations in a subset

Hello I am not a mathematician so please be understanding if my terminology is off. I will explain this using examples to be as clear as possible. I have a sequence of numbers [1,2,3,4,5,6,7,8] and I ...