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Questions tagged [permutations]

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

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Find the number of $6 \times 7$ matrices with entries $[0,1]$ such that their row and column sums are odd.

My attempt is quite handwaivy. But I think this has something to do with permutation matrices. I am absolutely new to this topic. can anyone throw any light on this solution? I know there are $2^{n\...
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permutation of {1,…,n}

Show for any permutation of $\{1,...,n\}$ = $ \{t_1,...,t_n\}$ there is g: $\{1,...,n\}$ $\longrightarrow$ $\{-1,1\}$ such that $|\sum_{i=1}^j g(i)| \leq 1$ and $|\sum_{i=1}^j g(t_i)| \leq 1$ for $j \...
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recursive relation on derangement of objects

Let $a_{n}$ represent the number of derangements of $n$ objects . If $a_{n+2}=p a_{n+1}+q a_{n}\;\forall n\in\mathbb{N}$ then what is $\displaystyle \frac{q}{p}$? What I have tried: I have used $$ ...
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What is the maximum number of subsets we can choose from a set of size 20 such that no two subsets have more than 2 common elemtns.

We have a set which has $20$ elements (eg. $\{1,2,3,...,20\}$). We want to choose the maximum number of non-empty subsets we can such that none of them has more than 2 common elements. for example $\{...
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2answers
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Distributing 5 distinct balls into 3 distinct boxes

Suppose $5$ distinct balls are distributed into $3$ distinct boxes such that each of the $5$ balls can get into any of the $3$ boxes. What is the Probability that exactly one box is empty. Also What ...
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Number of possible 5 by 6 matrices from 30 different numbers (without Repetition)

If I am trying to figure how many possible $5\times6$ matrices I can have from $30$ different numbers with no repetition allowed. What I am thinking of is that it will corresponds to $30! = 30\times29\...
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3answers
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How many permutations can be made of word “ASSASSIN” such that only $2$ Ss are together?

How many permutations can be made of word "ASSASSIN" such that only $2$ Ss are together? I have been doing by taking $2$ S together in a group and arranging them as $$\frac{7!}{2!2!}=1260$$ which is ...
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3answers
64 views

How to permute columns by pre-multiplying and rows by post-multiplying?

I was looking at Gilbert Strang's lectures on Linear Algebra and noticed that in lecture 2, Elimination with Matrices, around the 40nth minute he mentions that you can use the permutation matrix, $$P=...
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1answer
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Partial permutation of time sequence data that keep order of events

Suppose you have sequence S of N elements that are descending ordered by time. How many ways can you take K element subsets from S preserving time descending ordering? example for sequence S={A,B,C,D,...
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Find the total number of 20 digit codes that can be formed using the numbers {0,1,2,3,4}, such that consecutive digits have a difference of 1?

To start with an example of such a code can be: $34321210123212343210$ I have no clue how this property can be mathematically counted. I actually even have a short solution of this question which I ...
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1answer
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Number of combinations for a 4-character password with particular rules

We have a password with the following rules: 4 characters, no more, no less. Only normal alphabet characters (a...z) Only 1 uppercase character (but we don't know in which position). What steps ...
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1answer
31 views

Find the number of ways we can form words using each letter in the word $DISKRET$ exactly once, if certain words may not appear as subwords.

Good evening, any solutions or tips for this problem? Find the number of ways we can form words using each letter in the word $DISKRET$ exactly once, if none of the words $RET$, $SEK$ or $DIS$ may ...
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2answers
38 views

In a word containing $k$ A's, how many permutations place at least $n$ A's consecutively?

Suppose a word is $l$ letters long, and it contains $k$ A's. (The specific letter is irrelevant) Is there a general formula to count how many permutations contain at least $n$ consecutive A's? (Assume ...
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1answer
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How many different arrangements are possible? Combination & Permutations [closed]

Janet has 10 different books that she is going to put on her bookshelf. Of these, 4 are Chemistry books, 3 are Biology books, 2 are Statistics books, and 1 Physics book. Janet wants to arrange her ...
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1answer
27 views

How many 10-letter words can we find such that none of them are anagrams?

This question has two parts: a) How many anagrams does Mathematics have? This can be solved by counting the permutations of "mathematics" and removing the permutations of repeated letters. Thus, ...
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Prove that even permutations generate the endomorphisms of a hyperplane of $\mathbb{R}^5$

Let F be the hyperplane of equation $x_1+x_2+x_3+x_4+x_5=0$. For $g\in A_5$ (alternating group) and $x = (x_1, \cdots, x_5)\in\mathbb{R^5}$, we define $\rho(g)(x)=(x_{g(1)}, \cdots, x_{g(5)})$. $\rho$ ...
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Is there a term for permutations where the elements are optionally included?

The permutation for abc would be: abc acb bac bca cab cba But if the elements are optional: ...
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2answers
36 views

How many sequences of four numbers exist with these conditions?

First of all, this may seem very basic for you, but I've never been good at math, and I can't figure this out! I need to solve the number of sequences that exist under this conditions: All sequences ...
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1answer
58 views

Sitting on a bench and getting up without apologising

There's a long table and behind it - a long bench. The only way to sit and get up and leave from the bench is through one side. Behind the table, there are 14 students and they're writing an exam. ...
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1answer
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Given $N$ slots and $S$ objects to fill those slots, how many ways are there to fill the slots such that no two objects are adjacent.

Given $N$ slots and $S$ objects to fill those slots, how many ways are there to fill the slots such that no two objects are adjacent? I can't see a general pattern for this. If I take $N=7$ and $S = ...
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1answer
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Find permutations whose third power is known

I have to find permutations $a$ such that $a^3=(1 \ 2)(3 \ 4)(5 \ 6)(7 \ 8 \ 9 \ 10)$ and I have to find at least 3 solutions. So first I must find disjoint cycles. Those are: ...
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Have I calculated the correct EV of Fixed Points

Say we select a permutation $\sigma: [n] \to [n]$ from the permutation of all numbers $\{1,...,n\}$. $Q1$ Model the corresponding probability space. Next, define $X$ as an RV for the number of ...
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Selecting conditional states depending on previous states

I've seen a post which was started as a joke saying : "Well, Guess the code ?" (4 digit code) Apart from the joke , I was thinking , well , how many combinations do we have here , knowing that <...
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How may arrangement are there for letters of the word NOTABLE If each arrangement begins with a consonant and ends with a vowel? [closed]

I don’t understand the given problem: it’s quite difficult for me to analyze the question.
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Formula for r-Permutations of a Multiset

Suppose we have a multiset $M$, which contains $k$ distinct elements. Each element $x_i$ has multiplicity $n_i$ for each $i\in\Bbb{N}$ such that $0\le i<k$. $n$, the number of elements in $M$ ...
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Number of conjugates of $(12)(34)(56)(789) \in S_{10}$

Number of conjugates of $(12)(34)(56)(789) \in S_{10}$ This is how I calculated it and got $840$ as a result: $${10\choose 2}\cdot{\frac{2!}{2\cdot3!}}\cdot {8\choose 3}\cdot\frac{3!}{3}$$ What I ...
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1answer
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$f : S \rightarrow S$ is called cool, if for all elements $x$ of $S ,$ $f ( f ( f ( x ) ) ) = x$

Let $n \geq 1$ be an integer and consider a set $S$ consisting of $n$ numbers. $A$ function $f : S \rightarrow S$ is called cool, if for all elements $x$ of $S$ $f ( f ( f ( x ) ) ) = x$ Let $A _ { ...
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How to define closeness measure between a matrix and its permuted smoothed version.

I have a matrix $A$ with $n$ rows and $2$ columns. I want to smooth (let's say moving average) each column resulting in a smoothed matrix $B$ with $p$ rows and $2$ columns with $p \lt n$. I can smooth ...
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Why is the multiplication of permutations not commutative?

If the multiplication of disjoint cycles is commutative, and every permutation can be written as the product of disjoint cycles, then why is the multiplication of permutations not commutative?
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Arrangment Problem

Compute the number of different ways you can tile a 10x32 (height=10, width=32) rectangle using 1x2 and 1x3 tiles and complying to the following constraints: The rectangle should consist of 10 rows ...
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Transformation matrix of permutation

I need help with the following: Let $K$ be a field. Let $\sigma \in S_n$ be a permutation of numbers $1, \dots, n$ and let $f:K^n \rightarrow K^n, \begin{pmatrix} x^1 \\ \vdots \\ x^n \end{pmatrix} \...
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A question about permutation group and its subgroups

I have these two permutations of $S_{12}$: $\alpha =(1\;3\;5\;7\;9)(2\;4\;6\;8\;10)(11\;12)$ $\beta=(1\;6\;8\;10)(2\;3\;5\;7)(4\;9)(11\;12)$ I need to prove that if $G$ is a subgroup of $S_{12}$ ...
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1answer
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Permutations of UNIVERSAL

In how many of permutations of word UNIVERSAL, no two of the letters E, R, S occur together. I'm confused while proceeding with this one, I know total permutation is 9!.
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Is there a name for the quotient of the symmetric group by the finitary symmetric group?

The finitary symmetric group on a set $S$ is the group of permutations that only move a finite set points. That is: $$FSym(S) = \{\phi:\{s : s \in S, \phi(s) \neq s\}\text{ is finite}\}$$ This is a ...
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4 couples and 4 single people seated at 3 round tables

In how many ways can you seat the 12 people at 3 round tables such that: A) All couples are seated together. (the two members of each couple sit side-by-side) B) No couples sit together. I've ...
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How similar are permutation groups that are isomorphic as abstract groups?

Let's say that two permutation groups $P_1$ and $P_2$ are isomorphic as abstract groups, but not necessarily permutation isomorphic. How similar will $P_1$ and $P_2$ be, and how much structure will ...
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For $n>1$, let $H$ be the set of all products in $S_n$ of a multiple of four transpositions. Show $H=A_n$.

I'm reading "Contemporary Abstract Algebra (Eighth Edition)," by Gallian. This is Exercise 5.80 ibid. Answers that use only tools available in the textbook so far are preferred. For $n>1$, let $...
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2answers
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Ways to arrange $n\geq2$ people around a circular table, given two permanent seats.

How many ways to arrange $n\geq2$ people around a circular table, given two specific people who cannot stand next to each other? I've observed that when $n=2$ and $n=3$ there exists no way to arrange ...
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1answer
39 views

Counting 4-digit combinations such that the first digit is positive and even, second is prime, third is Fibonacci, and fourth is triangular

This seemed like a basic problem, but for some reason I can't figure it out: In a $4$-digit combination, the first digit has to be a positive even number, the second a prime number, the third a ...
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Terminology for sets which are permuted within themselves on repeated application of a function

Suppose that I have a function whose domain is equal to its codomain. One example of such a function is multiplication by a square matrix. In the matrix example, there often exist subspaces which are ...
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Show that $\chi(\cdot)$ is a non-trivial character on $(\mathbb{Z}/p\mathbb{Z})^{\times}$.

Let $G = \mathbb{Z}/p\mathbb{Z}$ with $p$ an odd prime. If $p \nmid a$ then multiplication by $a$ on the elements of G is bijective and therfore this is an permutation on G. Define $\chi(a)$ as the ...
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f interchanges $\alpha$ and $\beta$ but fixes $\gamma$ and $\delta$

$f=(\alpha \beta), \; g=(\beta \gamma), \; h=(\gamma \delta)$ Is there a way to solve this that avoids finding what the permutations in options are ?
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1answer
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Swapping elements to form a specific permutation - Formal Proof

Considering a permutation of [1, 2, ..., n], it is fairly obvious that on doing n/2 swaps we arrive at the permutation [n, n-1, ..., 1]. This can be achieved by swapping the first element with the ...
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1answer
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For which $n\in \Bbb N$ is $H_n:=\{\alpha^2\mid \alpha\in S_n\}\cong A_n?$

I'm reading "Contemporary Abstract Algebra," by Gallian. This is inspired by Exercise 5.73 and Exercise 5.74 ibid. I have a preference for answers using only the tools available in the textbook so ...
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1answer
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Combinations & Probability - Sport Club Teams Probability - Is my solution correct?

I'm revising permutations, combinations and probability for an upcoming exam and would really appreciate if someone could take a look at my procedure to solve this problem and let me know if it's ...
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Proof that composition of two permutations is again a permutation.

Permutations are symmetries of a (not necessarily finite) set $X$, often denoted as Sym(T). That is, a permutation $p: X\to X$ is a bijective map from a set $X$ to itself. I wish to prove the ...
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Proof for the determinant of a Cauchy Matrix

I want to proof the formula for the determinant of a Cauchy Matrix without recurring to matrix manipulation, but by directly applying the definition of the determinant. That is, given two sequences ...
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0answers
19 views

Show $\nexists x\in S_7$ with $x^2=(1234)$ but there is at least two $x\in S_7$ with $x^3=(1234)$. [duplicate]

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 5.48 ibid. and I want to answer the question using the tools available in the textbook so far. (A free copy of the book is ...
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Does $A_8$ contain an element of order 26?

The question is does $A_8$ have an element of order 26? My knowledge on alternating groups is very limited. I understand that the order of $A_8$ is $\frac{8!}{2}$ but that is essentially all. My one ...
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1answer
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Equivalence relation of transposition has equivalence groups of all same size.

Let $G$ be a transitive subgroup of the permutation group $S_d.$ Now define an equivalence relation on $S = \{1, \ldots, d\}$ by $i \sim j$ iff the transposition $(ij)$ exists in $G.$ How do I show ...