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

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Counting with permutations and counting ignoring permutations

I am given this problem: This problem was given to me in my computer science class but it has to do with permutation and I want to understand it mathematically first. let $c(n)$ be the number of ...
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1answer
67 views

Four Letter-envelop problem

A secretary writes four letters and the corresponding addresses on envelopes. If he inserts the letters in the envelopes at random irrespective of the addresses, (i) find the probability that only one ...
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37 views

Number of visible elements in a permutation

The following problem occurred to me the other day, and I've played around with a bit but can't seem to find a good solution: Consider a permutation $\pi$ of $\{1, 2,\ldots ,n\}$. For every positive ...
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18 views

probability of vector in column span

Consider we have a fixed matrix M of size a$\times$ 2b (Let us look at M=[$M_1$ $M_2$] where matrices $M_1$,$M_2$ are of size a$\times $b) and a vector $v$ of dimension a. Is there any way that I can ...
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1answer
126 views

Outer automorphisms of the infinite symmetric group

Denote by S$_\infty$ the group of permutations of $\mathbb N$. Question: Does there exist an outer automorphism of S$_\infty$, and if so, can one be exhibited? Does this depend on the continuum ...
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51 views

Generators of symmetric group $S_n$ [duplicate]

How can you prove that $S_n$ is generated by $(1\space 2)$ and $(1\space 2\space 3\space ... \space n))$ for $n\geq 2$?
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1answer
15 views

Finding cycles with set of permutations

Let $\alpha = (\alpha_1 \, \alpha_2 \, \ldots \, \alpha_s)$ be a cycle, for positive integers $\alpha_1 , \alpha_2 , \ldots , \alpha_s$. Let $\pi$ be any permutation. Show that $\pi \alpha \pi^{-1}$ ...
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17 views

the number of permutation $p$ of $A$ such that $\forall i\forall x\in A_i \,\,\, p(x)\notin A_i $

Given a collection of finite sets $(A_i)_{i=1,...,n}$, Let $A=A_1\cup\cdots \cup A_n$, my question is : What is the number of permutation $p$ of $A$ such that $\forall i=1,\cdots,n$ we have ...
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1answer
49 views

Does the product of all conjugates of some subgroup is independent of the order?

Let $G$ be a finite group and $A \le G$. Let $A^G = \{ A_1, A_2, \ldots, A_n \}$ be all the conjugates of $A$, i.e. each $A_i$ equals $A^g$ for some $g \in G$. Then I want to show that $$ A_1 A_2 ...
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1answer
23 views

permutations of 10 objects in a subset contains similar elements

A board that is divided into 15 different places, and we want to place 10 components on this board such that each component is placed in a section; knowing that those components are divided into 4 ...
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3answers
65 views

Simple Probability Question about Combinations

If someone could please point me in the right direction on these. I get lost on how to think about them. In a game there are four holes with values 0, 1, 2, and 4. You are given 6 balls to shoot into ...
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1answer
23 views

Double Check Probability for Permutation

I have to find the sample space and a few probabilities here and I am wondering about if I am going down the right track for these. If I am incorrect, then please point me in the right direction, but ...
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2answers
32 views

Question about permutations: How to show $\sigma(P)=(-1)^{\imath(\sigma)}P$?

A permutation of a finite set $X$ is any bijection from $X$ to $X$. We denote by $S(X)$ the set of all permutations of $X$. If $I_n:=\{1, \ldots, n\}$ we write $S_n$ instead of $S(I_n)$. Define ...
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2answers
30 views

Parity of permutation example

I know the definition of parity of permutation. But what does that look like in examples? For example, if the number of permutations is odd, then the sign of permutation in $-1$. What does this mean? ...
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2answers
38 views

24 possible combinations?

I'm terrible at math. My goal was to try and create $24$ total combinations using a horizontal bar, and a horizontal bar with one break. So that's two bar types. How can I write out the math that ...
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1answer
20 views

How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?

Question: 2 Spaniels, 2 retrievers and 3 poodles go through to the final. They are placed in a line. How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?
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1answer
50 views

Number of possible passwords - Google APAC Test

I am trying to solve this problem - Google apac test - Password Attacker . Problem summary: Using $M$ distinct characters, what is the number of ways of making a password of length $N$, such that ...
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1answer
27 views

Example of an elementary permutation

Can someone please give an example of an elementary permutation? The book says that every permutation can be written as a composite of elementary permutations. Can someone please give an example? ...
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27 views

Count number of trees

Given an array with n elements which is the pre-order traversal of a tree. How many max-heap will have the same pre-order traversal?
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1answer
83 views

Arrangement of integers in a row such that the sum of every two adjacent numbers is a perfect square.

Inspired by this interesting question and in order to solve an old problem, I have the following question: Can we construct a strictly increasing sequence $(N_i)_{i\in \mathbb{N}}$, such that for ...
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1answer
46 views

Murder mystery permutation problem

In exploring a hypothetical situation, I ran across this problem and I'm curious to know the answer, but math's not really my forte. You have a pool of 15 people. Between these 15 people there will ...
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2answers
61 views

How to find the number of strings of length N that can be formed by using the characters A, B, and C only that do not have “ABC” as a substring?

A, B, and C can be used any number of times in the string. This problem appeared in a programming contest which is already over. http://www.codechef.com/problems/CDSW152
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1answer
29 views

What kind of mathematical approach can you use to find all non-repeated combinations?

At first glance I thought this was a non-repeated combination or permutation, but those use a set length. So, I guessed this might be a partition of a positive integer, but it's not looking like that. ...
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1answer
49 views

An Example for a Graph with the Quaternion Group as Automorphism Group

I am reading "Graphs of Degree Three with given Abstract Group" (by Robert Frucht) where the author describes (somewhat tedious) algorithms to construct suitable graphs starting from a given group. I ...
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How to calculate total results of combinations of letters

I am programmer and have developed an algorithm to run a processor intensive function on all the permutations of 2 letters (X and O) when we define how many X's and O's there will be. For example, I ...
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34 views

Structure of the semidirect product decomposition

I'm looking at a complicated group that involves many semidirect products, and I realized that I have a fundamental confusion about how to use the structure of a semidirect product decomposition of a ...
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1answer
48 views

Possible permutations of a grid

I hope this is the correct place to post this, as I don’t study maths. But I do need help calculating the possible permutations of a grid based game I’m currently programming. This isn’t to help out ...
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1answer
39 views

Find Number of combinations possible

There are two letters "X" and "Y".A String of length N needs to be formed using those two letters How many number of combinations that can be possible where N should start with "Y" and no two or more ...
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31 views

Number of ways to color N objects in X colors where there is at least one object of each color.

What is the number of way to color $N$ objects in $X$ colors, where there is at least 1 object of each color?
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1answer
104 views

Combination or Permutation

I am searching for the number of uniques ways to paint an icosahedron. However, my understanding of mathematics is quite limited in the field of combination and permutation. I have searched through ...
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0answers
15 views

Generating uniform permutations by a particular method

Let $A$ be a uniformly random permutation of the numbers $\{1,2,\cdots,n\}$. I want to generate a uniformly random permutation from $A$ on the numbers $\{1,2,\cdots,n,n+1,\cdots,n+m\}$. In other ...
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1answer
49 views

How many cycles of length $k$ in $S_n$?

In the symmetrical group $S_n$, how many cycles of length $k$ can we form? After some research I am tempted to say $\frac{n!}k$ but I am not sure.
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Dependent permutations, a question.

I cant seem to find anything on the internet on this subject , and the professor did not explain it too well, in short the following is unclear to me how is $$(1 3 4)(236)=(24136)$$and ...
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1answer
38 views

Symmetries of the regular hexagon

Q- Let G be the group of the symmetries of the regular hexagon. List the elements of G (there are 12 of them), then write the table of G. So for the listing the elements of G, they want it like this: ...
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43 views

Integrating over a symmetric-group function (elements being permutations)

I would like to integrate a permutation of a function. Namely I have the following: $\sum_{\sigma, \sigma'\in S_{n+1}}\int_{-A}^A dz_1dz_2 ... dz_{n+1} ...
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31 views

Ball-of-wacks combinations

The six-color version of the ball-of-wacks consists of thirty rhomboidal pieces, which can be combined to form a rhombic triacontahedron. There are six colors, each with five pieces. One challenge ...
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2answers
30 views

Illegal permutations give a nonzero answer

I am told that a random variable can take a value of $+1$ or $-1$. I am given the total number of times the random variable is counted, $N$, and the sum of the random variables, $n$, and asked to find ...
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1answer
48 views

List the elements of the cyclic subgroup of $S_6$

List the elements of the cyclic subgroup of $S_6$ generated by: \begin{smallmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 1 & 6 & 5 \end{smallmatrix} ...
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1answer
62 views

How many three digit number can be formed?

Question: (a) How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once? (b) How many of these are odd numbers? (c) How many are greater ...
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Birkhof representation of a stochastic matrix

From Birkhof Theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices although this representation might not be unique. I have the ...
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1answer
34 views

no of possible ways [duplicate]

we have to build a houses on $m$ plots, such that no two consecutive plots exist on which it is allowed to build house calculate the number of possible ways of assigning free plots to buildings ...
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1answer
45 views

The order of a $k$-cycle in $S_n$ is $k$.

Here's what I have right now: The order of a $k$-cycle in $S_n$ is $k$. Proof. Let $\sigma$ represent the $k$-cycle $$\sigma=(x_1 \ x_2 \ \cdots \ x_k)$$with distinct elements $x_i$. Note that the ...
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1answer
23 views

How to solve this statistics problem? [closed]

Can you find the sum of all numbers that can be formed with the digits $2, 3, 4$ and $5$ taken all at a time? (So its like you sum up the number from 1st digit to 4th digit) I'm learning now about ...
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2answers
24 views

Permutation's decomposition into transpositions

Transposition is a cycle with 2 elements. Any permutation can be decomposed into a product of transpositions. For example, for permutation $\begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 3 ...
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1answer
42 views

Maximize $a_1^{a_2^{\ldots^{a_n}}}$, where $(a_1,a_2,\ldots,a_n)$ is a permutation of $(b_1,b_2,\ldots,b_n)$

You are given a tuple of integers $B=(b_1,b_2,\ldots,b_n)$. Find $(a_1,a_2,\ldots,a_n)$ - a permutation of $(b_1,b_2,\ldots,b_n)$ - that maximizes $a_1^{a_2^{\ldots^{a_n}}}$. For example - If ...
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49 views

Permutations Without Repetitions

Given the set [A,B,C,D] how many distinct ways can I order all four of the members of the set? I see distinct, as a unique set, therefore [A,B,C,D] and [D,C,B,A] ...
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1answer
81 views

How to calculate combinations of multiple variables which can assume multiple values

I have 3 variables (A,B,C); each variable can assume 3 different values (1,2,3) . I want to calculate ho many combinations there are which follow this rule: let's fix A1, then cycle on all the others ...
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3answers
283 views

How many ways to reach a given tennis-score?

Let's say a tennis player wins a set with a game score of 6-3. In how many different ways can we reach this score? Assuming H means the home-player won the game and A means the away-player won the ...
4
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2answers
76 views

Number of Words with two letters $a$ and $b$.

Given $N$ and $M$, find the number of $N$ letter words consisting of only $a$ or $b$, where $b$ must not be consecutive for more than or equal to $M$ times. Example: if $N=3$ and $M=2$, then all the ...
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1answer
29 views

Combinations - Permutations

Find the number of ways in which 5 books can be distributed between three people A,B and C, if the books are a)indistinguishable, b)all different. a) $\displaystyle \frac{5!}{3!(5-3)!} = 10$ b)$ ...