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

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1answer
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Order of a permutation using its cycle decomposition

If $A=\{1,2,...,n\}$, $\Omega _A$ is the set of all permutations over $A$, $S_n=(\Omega _A, \circ)$, then for any $\sigma \in \Omega _A$, the order $m$ of $\sigma$ (Smallest $m \in \mathbb{N}$ for ...
6
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1answer
67 views

Why is the parity of a permutation an important concept?

In Pinter's A Book Of Abstract Algebra, the author states that: A number of great theorems of mathematics depend for their proof (at that crucial step when the razor of logic makes its decisive ...
3
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4answers
45 views

Is there a way to assign a number to a combination without finding and numbering every combination?

Imagine I have 4 letters. Is there some algorithm that produces "abcd" -> 1 "bacd" -> 2 "bcad" -> 3 ... etc without finding and numbering every single combination? My goal is to get a number from 1 to ...
-1
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4answers
55 views

I need a hint to prove that the number of ways of arranging N distinct items on a circle is (N-1)! [closed]

Can anyone explain me a proof for arranging $N$ distinct items in a circle is $(N-1)!$ I need as early as possible. Could you please guys Elaborate the procedure by in a step manner which is very ...
0
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0answers
19 views

3 permutations of 14 letter sequence with repeatnig letters

I have a set $S=\{u,n,6a's,2l's, 2k's, 2s'\}$, $|S|=14$. How do I compute number of 3-letter sequence. I approached this by giving myself a smaller example of a set $\{n,3a's,2l's\}$, I split that ...
1
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5answers
163 views

How many permutations of {1,2,…, 9} are there that do not start or end with an even number?

How many permutations of $$1,2,..., 9$$ are there that do not start or end with an even number? This is my attempt Condition 1 [Starts with even] => $$4 * 8!$$ Condition 2 [Ends with even] => $$4 * ...
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2answers
36 views

Couple of Counting (how many ways) questions.

1.If I have a group of 10 seats reserved for people, and there are n=>10 total people, how many ways are there to choose who gets the 10 seats? for ex:If there was a definite number of people lets ...
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0answers
20 views

generating locally random permutations

I have an intuitive notion of 'local randomness' that I am trying to make precise and understandable, and I am running into a bunch of problems. A quick web search failed to find anything relevant in ...
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2answers
44 views

Distributing $7$ books to $2$ persons such that each person gets at least $1$ book

In how many ways can $7$ different books be distributed to $2$ persons if each person gets at least $1$ book? I did my calculations and my answer is $126$, but the answer stated is $216$.
0
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1answer
28 views

How to prove that determinant with permutation symbols

How to prove that $$\varepsilon_{ijk}a_{i\ell}a_{jm}a_{kn} = \det[a]\epsilon_{\ell mn}$$ I'm trying to solve this problem with permutation symbol but i can't solve it Help me,please. Thank you ...
1
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1answer
50 views

Partitions of n with certain conditions

Let $p$ be prime and $n$ be any integer. Suppose $t=(n^{a_n}, \dots, 2^{a_2}, 1^{a_1}) \vdash n$, (i.e. $t$ is a partition of $n$, where we group repeated integers, so, for example, $2^{a_2}$ means ...
3
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0answers
35 views

Number of ways to order a list of permutations by swapping

I want to solve a problem and I have no idea how or where to begin. I don't even know if it's possible to solve. I tried to find any clues in books about discrete maths but I didn't find anything that ...
2
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2answers
66 views

What is the probability no slots contain more than two balls given I am trying to sort 5 balls into 6 slots?

I am having a difficult time understanding how to approach this problem. Suppose I have $6$ total slots and $5$ balls. Now, I assign the balls at random to the slots. What is the probability that no ...
0
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0answers
33 views

Shuffle the Kronecker Product

Suppose $D$ is a orthogonal matrix of size $N\times N$, and we have \begin{equation} S = D F D^{T}, \end{equation} where $D^T$ means the transformation of $D$. Using Kronecker product, we can rewrite ...
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0answers
33 views

A conjugation of an operator, which commutes with all permutations, still commutes with all permutations

Assume $v:H^{\otimes m}\to H^{\otimes m}$ is a linear operator on the $n^{\text{th}}$ tensor power of a vector space $H$. For each permutation $p$ on the $m$-element set define the linear operator ...
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1answer
35 views

The anti-symmetrization and simetrization operators are mutually orthogonal

For each vector $x=(x_1,\dots,x_n)$ of an $n$-dimensional vector space $V$, and for each permutation $s$ of the symmetric group on the $n$-element set $S_n$, put $s(x)=(x_{s(1)},\dots,x_{s(n)})$. Then ...
0
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1answer
25 views

Word combinations Bamboozled?

I have a list of 1,626 random words. How can I work out the total number of combinations abiding to a limit of 12 words per thing ? E.g dog fish cat whale shark snake spider eagle nine dog clam ray ...
2
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0answers
43 views

What is the probability of winning.

$2$ teams of $5$ players each play a game against each other in pairs. Find the probability that no team wins all the games and no team loses all the games. I tried to solve and determined that there ...
1
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3answers
69 views

Number combination with groups and the smallest repetition possible

I need to create a equation to assign a number of phrases (variable A) to a a defined number of groups (variable B) and repeat these assignment each day, and repeat this operation along of time with ...
1
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2answers
65 views

How many $6$ digit numbers can be formed with the numbers $0, 1, 2$ if the number must contain at least one $0$?

So the question is: How many $6$ digit numbers can be formed with the numbers $0, 1, 2$ if the number must contain at least one $0$? I have searched and found some similar questions so I hope this ...
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2answers
61 views

Permutation Group-$S_{10}$

How many elements of order $30$ are there in the symmetric group $S_{10}$? I worked out and got $10500$ - using Computations and adjusting each individual cycle's position ( $2-3-5$ , $3-2-5$ etc). ...
3
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0answers
86 views

Circular permutations - $n$ sitting at a round table without repeating neighbors

I hope this isn't a duplicate - the problem is to find the number of ways of sitting $n$ people (who initially were sitting in the order $1, 2, \dots,n$, with $1$ and $n$ being neighbors) at a round ...
0
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1answer
47 views

Combinatorics question - How many different ways to change sitting order

Six children ($a$ through $f$) are playing on a carousel with 6 seats such that $a$ is sitting in front of $d$, $b$ is sitting in front of $e$, and $c$ is sitting in front of $f$. How many ways are ...
0
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1answer
23 views

What's the right way to look at this permutations problem?

I'm having trouble with this problem. It seems very simple. Here it is exactly: Obtain the number of three–letter permutations possible for the group of letters shown. S, E, V, E, N ...
0
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1answer
58 views

How many functions are not one-to-one?

I am having some trouble starting this question. If we have two sets. Set A of size m where m≥1 and set B of size n where n≥1. How many of the functions f : A→B are not one-to-one? I know that the ...
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1answer
44 views

Prove a polynomial in Fq is a permutation polynomial of Fqn with a necessary and sufficient condition

P.S This is the best Math Expression I can edit. I am real shameful, where can I find the introduction of typing in this webset? thank you Exercise7.13 Let\[f\left( x \right) = \sum\limits_{i = ...
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1answer
92 views

How many matrix colorings are possible?

We have a small square matrix having size up to $8$. And we have a large number of colors up to $10^6$. In how many ways we can color the matrix so that all the same color cells are not adjacent? ...
0
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2answers
26 views

Ternary in a 10-digit string

I encounter an interesting question and can't seem to form a logic for solving it. We need to form a 10-digit string using 0, 1, or 2 (ternary string). There should be exactly 3 0's. In total, how ...
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1answer
26 views

Convex Hull of cyclic Permutations

It is known that the convex hull of permutation matrices yields exactly the stochastic matrices. I am interested in the convex hull of cyclic permutation matrices. Trivially this is a subset of the ...
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3answers
88 views

Generalization of permutation matrix

For integers $n$ and $k$, I am interested in $n\times n$ matrices with exactly $k$ non-zero entries in each row and each column. The case $k=1$ corresponds to (generalized) permutation matrices. In ...
3
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0answers
59 views

Probability/selection problem

Assume that we have $N$ items of $M$ distinct types in a closed bag. We also have $K$ bowls $(K \leq M)$ that can hold only items of same type. In the beginning bowls are empty. And bowls can hold a ...
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2answers
36 views

Combination on jury selection

$20$ women (including Alice and Betty) and $12$ men show up for jury duty. In how many ways can you select a jury of at least $5$ women and at least $5$ men if one of Alice or Betty must be selected, ...
1
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3answers
57 views

Permutations for paths

What is the counting sequence for paths from $$(0,0)\text{ to }(n,n)$$ where $$2n$$ is the size of the path (number of steps) and n can vary over all nonnegative integers. I don't know how to ...
0
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1answer
49 views

Four Children Combination Problem

This is a problem that has haunted me for some time There is two family. First Family : ...
0
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3answers
124 views

Finding the probability that three friends get into the same group [closed]

I am stuck with the following problem: Students of a school are divided into $\,4\,$ groups. What is the probability that three friends get into the same group ? The options are : ...
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1answer
23 views

Number of orders and combinations

I have just done these two questions and I have answers for them but I am not sure if they are correct. A jazz band is to give one concert in each of nine selected cities. Calculate the total ...
0
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2answers
46 views

Permutation question

I'm trying to solve this question In a photo there are three families six Greens, four Browns and seven Grays arranged in a row. The Browns have had an argument so no Brown will stand next to another ...
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8answers
2k views

Calculating the number of possible paths through some squares

I'm prepping for the GRE. Would appreciate if someone could explain the right way to solve this problem. It seems simple to me but the site where I found this problem says I'm wrong but doesn't ...
0
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1answer
60 views

Finding the number of different relations and functions

This must be a very stupid question. Let set $A=\lbrace{a,b\rbrace}$ and $B=\lbrace{1,2,3\rbrace}$. The total number of relations from $A$ to $B$ is $6$. We can calculate this as a has $3$ choices and ...
2
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1answer
30 views

How to convert a permutation to permutation polynomial?

Let Fq be the finite field with q elements, where q is a prime power. A permutation on Fq is a bijection from Fq to itself. Let Fq[x] be the ring of polynomials in a single indeterminate x over Fq. A ...
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2answers
79 views

Example of a simple graph isomorphic to a permutation group.

I'm taking a first course in graph theory this semester and I'm working trough Graph Theory with Applications by J.A. Bonday and U.S.R. Murty. I can't find an answer to question 1.2.12(f): (a) ...
2
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1answer
44 views

Taking square root of cycle permutations

Let $\alpha$ and $\beta$ be permutation cycles of $\{1,2,\ldots,n\}$ such that $\alpha^2=\beta^2$ Can we conclude that $\alpha=\beta$, if (a) $\alpha,\beta$ are odd? (b) $\alpha,\beta$ ...
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1answer
62 views

Permutation that fix elements within set is subgroup?

(a) Let $A$ be a finite set, and $B\subseteq A$. Let $G$ be the subset of $S_A$ (permutations of $A$) consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove ...
0
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1answer
19 views

$11$ matches are to be played,Each having $3$ distinct outcome,

$11$ matches are to be played,Each having $3$ distinct outcome, in how many ways one can predict the outcomes such that $6$ outcomes turn out to be correct? My thought $11C_{6}\times 3^5$ am I ...
5
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1answer
108 views

Intuition - Identities with 2-Cycles and 3-Cycles - Mulholland p. 69, 86 - Fraleigh p. 90

Jamie Mulholland p. 69 Theorem 6.1 or Fraleigh p. 90 Corollary 9.12 Any permutation of a finite set of at least two elements is a product of 2-cycles. $1. (a_1, a_2, ···,a_n)= (a_1, a_n)(a_1, ...
0
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1answer
33 views

Filling the grid (combinatorics). [closed]

We have 4 O(circles) and 5 X(crosses) available and we have to fill the gird with them. given the following condition. Find in how many ways this would be possible with explanation. Figure states ...
3
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3answers
58 views

permutation and combination confusion

Can you point out where am I going wrong ? I wanted to select 5 men from 7 men . This can be done very easily as 7C5 ways = 21ways, but I was confused as to why the following calculation didn't work ...
2
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0answers
53 views

How many combinations does Android pattern have?

Rules- 1) At-least 4 and at-max 9 dots must be connected. 2) There can be no jumps 3) Once a dot is crossed, you can jump over it.
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1answer
33 views

Permutations from a set

Part of my problem is I can't figure out which question answers my problem. I'm not so familiar with the kind of math lingo that I know how to ask this question, so I'm gonna bumble my way through ...
0
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2answers
196 views

How do I find the maximum number of knights on a chess board?

I came across this problem and after thinking a lot I could not get any idea how to calculate it. Please suggest to me the right way to calculate it. Given a position where a knight is placed on ...