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

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Fundamental confusion on set theory and permutations

I am confused on the following: A set does not have any order. Now I read that a permutation is a bijection of a set. But doesn't this imply an order? I mean a bijection is a one-to-one function from ...
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1answer
52 views

Maximum winner matches

N players take part in tennis championship. In every match loser is out. Two players can play a game if in that moment the difference of played games of that two players is not more than 1. They are ...
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64 views

Is there away to swap adjacent items in a sequence, such that each permutation occurs once?

Let's us say you have a finite sequence of things. Some are identical, some distinct. For example: $$\langle 2,5,7,2,3\rangle$$ Now, under what conditions can each permutation of a sequence be ...
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2answers
124 views

How many permutations of all the letters in the word ARMADILLO begin with letter A?

I know that there are 9 total letters and there are three A`s and two L`s. Is the answer just $9!/(3!*2!)$? Thanks for any help.
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144 views

Do cyclic permutations of rows and column entries generate all permutations?

Background: I am interested in the group of permutations of the entries of a general $m\times n$ matrix. In particular, I am interested in (1) interesting sets of simple generators for this group ...
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3answers
260 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
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1answer
371 views

Collinear points and straight lines and triangles

There are 20 conplanar points of which 5 are collinear. How many straight line segments and how many triangles can be made using these points? I understand it belongs to combination. But there are ...
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2answers
53 views

If $F$ is a field, show the following function is a permutation

Let $F$ be a field. Show that the function $a\rightarrow a^{-1}$ is a permutation of $F\{0_F\}$ So I know that if it is indeed a permutation, then it is one-to-one and onto. Also, For every $a$,$b$ ...
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0answers
45 views

Possible number of permutations

I've generally been seeing the formula nPr = n!/(n-r)! to calculate the number of permutations. But this is the case only when elements cannot be repeated, am I correct? If elements can be repeated, ...
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38 views

How many different sequences in this event?

There are $10$ students in the class. Four of these students will be selected randomly to participate in the math contest. Then, how many different sequences can four students be selected? I thought ...
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1answer
25 views

Suppose we have a chair with $n$ legs and it stands with all legs irrespective of floor's quality i.e smoothness,evenness, then what is $n$?

Suppose we have a chair with $n$ legs and it stands with all legs irrespective of floor's quality i.e smoothness,evenness, then what is $n$? 1)2 2)3 3)4 4)5 I have no clue what they want to say, ...
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3answers
43 views

Generalize the number of total overall hops possible in a line network

Suppose I have a network of computers arranged in a line, like in the image I made below. I want to know the total number of hops possible. For example, $A$ gets to $B$ in $1$ hop, to $C$ in $2$ and ...
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1answer
18 views

How many configurations

If I have twelve objects, each of which can be in one of three states (say on, off, on/off), how many ways can the group of twelve be configured -- by changing the state of each object?
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1answer
70 views

Number of k-colorings as a fraction of all possible ways to color a graph

I have a graph with $n$ verices, and I want to compute the number of ways to color the graph (with no adjacent vertices having the same color) using anywhere between $1$ and $n$ colors. This number ...
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2answers
44 views

Find the number of different teams that can be selected if the team contains all $4$ women?

A team of $6$ people is to be selected from $8$ men and $4$ women. Find the number of different teams that can be selected if: (i) there are no restrictions, (I solved this using ${n\choose ...
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2answers
91 views

Find how many of these $4$-digit numbers are even. [closed]

(a) (i) Find how many different $4$-digit numbers can be formed from the digits $1, 3, 5, 6, 8$ and $9$ if each digit may be used only once. I did this; the answer is $360$; I used ...
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1answer
50 views

$PGL_d(F)$ is 2-transitive but not 3-transitive if $d > 2$

An exercise asks to prove that: If $d > 2$, then the projective general linear group $PGL_d(F)$ of dimension $d$ over a field $F$ is 2-transitive but not 3-transitive on the set of points of the ...
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0answers
65 views

password attempts

The password must have a length of 8 and be composed of lower case letters and/or numbers (letters and numbers can be repeated). Suppose someone wants to access your online bank account. ...
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1answer
43 views

Permutation that gives a sequence of non-negative partial sums

Given a sequence $(a_i)_1^n$ of real numbers that sum to $0$. There are $n$ circular permutations. $\begin{array}{}\sigma_1 = &(a_1 &a_2 &\cdots &a_{n-1} &a_n)\\ \sigma_2 = ...
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1answer
49 views

Understanding the structure of a module over a group algebra

Suppose one has a permutation group $G$ acting on the set $[n] = \{1, 2, \ldots, n\}$, which extends naturally for any field $F$ to a $FG$-module structure on the set $F[n]^k$ of formal $F$-linear ...
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1answer
40 views

number of possible combinations of events

I am not a mathematician but I'm giving a lecture on the scope of a number of people in a given group to interact. There are 40 people in an affinity group Each person has 10 unique items of ...
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1answer
87 views

How to calculate the HHG (Heller Heller Gorfine) correlation

HHG (A consistent multivariate test of association based on ranks of distances) is introduced in: Heller, R., Heller, Y., & Gorfine, M. (2012b). A consistent multivariate test of association ...
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3answers
59 views

Permutations with repetitions - which is k and which is n?

I am learning permutations with repetitions, and working with the formula that $P(n,k)=n^k$. I understand the logic that with repetitions, we multiply $n$ by itself $k$ times. But in different ...
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3answers
55 views

Permutation Formula Question

The problem that I am having trouble figuring out is: In how many ways can give five men and five girls be seated at a round table so that each girl is between two men? I know that the formula for ...
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1answer
43 views

probability of finding a small sequence within a larger sequence

I'm wondering how to define the probability of a long string LS (using 26 letters alphabet) to contain a smaller string ss. Right know I have something like this. Number of LS containing ss: 26^( ...
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2answers
42 views

3 letter combination

Please excuse me if this question is too juvenile for this forum. Actually weirdly I cant figure this out. I have 3 letters $A, B, C$ I need a formula that gives me all the possible arrangements for ...
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2answers
44 views

combinations and permutations - choosing when there's a limit

A kid can choose 7 out of 12 donuts to eat. How many ways can he do this if he must choose at exactly 3 of the first 5? Similarly, how many combinations are there if he must choose at least 3 of the ...
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2answers
118 views

Combinations and permutations when separating into groups

If I have 30 people and I want to form them into 3 groups. One of size 10, one of size 5 and one of size 15. How many ways can I do this? Similarly what if I have 15 boys and 15 girls and the 10 ...
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1answer
93 views

Write the permutation as product of cycles

Let the permutation $$\sigma=\bigl(\begin{smallmatrix} 1 & 2 &3 &4 &5 &6 & 7 &8 & 9 & 10\\ 3& 4 & 5 &6 & 7 & 2 & 1 & 10 & 9 ...
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430 views

combination GRE problem 25

An appliance's model number consists of three alphanumeric characters. The first character must be one of 24 permissible letters of the alphabet. The next character is numeric, a digit from 1 to 9. ...
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84 views

Asymmetry of random graphs

By a well known result of Pólya we know that the number $g_n$ of isomorphism classes of simple graphs on $n$ vertices is asymptotically equivalent to $\frac{2^{\binom{n}{2}}}{n!}$. In this paper the ...
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2answers
82 views

List vs Permutation representation.

If we have a list xx:=[3,7,17,16,15]; and I want to convert this list into permutation like xx:=(3,7,17,16,15); I used PermList(xx) for this but it doesn't convert it in the required form. How can ...
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0answers
49 views

Find k such that, $\displaystyle\frac{n^{\underline k}}{n^k}<a$

How to find k here $\displaystyle\frac{n^{\underline k}}{n^k}<a$ ? With, $n^{\underline k}=n\cdot(n-1)\cdot...(n-k+1)$ and $(n>k,\ a>0)$ Of course if $k\approx n/2$ the inequality ...
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60 views

show permutation graph is a perfect graph

I need to prove that permutation graph is a perfect graph. I've shown that for any $i<j$, $\{i,j\}$ connected iff $f(i)>f(j)$. In addition, I know that the complement of a permutation graph is a ...
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1answer
186 views

Dividing a set of n elements into k disjoint subsets.

I have been able to do the 1st part. I have not been able to prove the 2nd part. My attempt to the solution :- I took $k$ groups $ a_1, a_2, a_3…, a_k $ Let $a_1$ group has $b_1$ similarly so ...
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1answer
100 views

Number of subset-permutations for a set

Suppose you have a set {a, b, c}. There are 3! = 6 permutations. Subsets {a, b}, ...
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1answer
327 views

What is the meaning of nCk X nPk?

I am trying to understand the bulls and cows document, Page 6 , equivalences . Can someone please tell me what author means when he says nCk x nPk like 4P0 X 4C0 , 4P1 X 4C1 ?
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2answers
232 views

|x|+|y|+|z|=15. How many integer solutions do exist?

I tried to truncate the problem to $|x|+|y|=15$, which is giving me the answer to be $60$ solutions. However, I am trying to apply beggar method and everything shatters. How should I proceed? Can I ...
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2answers
119 views

Probability of boys ahead girls

There are three boys and two girls in a Queue. What is the Probability that number of boys ahead of every girl is at least one more than number of girls ahead of her. My Try: Let $B1$,$B2$,$B3$ are ...
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51 views

Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
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2answers
94 views

A Circular permutation problem

i've this problem that's making me crazy. I've 12 ball, 4 of these are red, the others are white. What's the probability to obtain a circular sequences of balls in which red balls are not adjacent? ...
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1answer
285 views

Mean and Variance of Correct Answers for various ways of Multiple Choice Questions

I picked this question out of the blue while thinking about multiple choice questions. Consider I set, say $8$ multiple choice questions. Now, I want to find out, out of the three methods below, what ...
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2answers
139 views

Burnside's lemma - show that there are just five necklaces

Show that there are just five different necklaces which can be constructed from five white beads and three black beads. Sketch them. The lemma tells us that The number of orbits of G on X ...
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2answers
74 views

Seeking a derangement $S: \{0,1\}^4 \to \{0,1\}^4$ for which changing one bit in the input always changes at least two bits in the output

Kindly asking for any hints about the following question: Define a function $ S : \{0,1\}^4 \to \{0,1\}^4$ with this conditions: 1- For any $x$, $\ S(x) \neq x $ 2- for any $x \neq y$, $\ S(x) \neq ...
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1answer
127 views

Show: group $G$ has one orbit on $ X$, stabilizer of $z$ is $3$…

If $X$ denotes the set of corners of a cube and let $G$ denote the group of permutations of $X$ which correspond to rotations of the cube. (i) $G$ has just one orbit (ii) if ...
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1answer
39 views

Find the permutation

This is part of an exercise I did on an assignment but I am having trouble remembering how to complete the exercise (even though I got full marks on my assignment). Let $P_1=(3\,4\,1\,2\,5), ...
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1answer
72 views

Theorem in finite fields fails in my example

I need to understand the following theorem, so i did an example. But i realized that i don't get everything in the finite field theory. can somebody check the example and say where the mistake is? ...
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1answer
41 views

A common combinatorics problem

Suppose the final result of a football match is 5−4, the home team winning. If the home team scored first and kept the lead until the end, in how many different orders could the goals have been scored? I ...
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1answer
109 views

How to get all permutations of a variable-length word

I need to find all permutations of a set of letters (word) with following parameters: Word lengths $\ell = [1, 20]$ Alphabet $A = \{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t\} \Rightarrow \lvert ...
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1answer
242 views

About generator of symmetric group $S_n$

I am reading this link . In Theorem $2.7 $, it is mentioned that for $n\geq 3$ except for $n = 5, 6, 8$, symmetric group $S_n$ is generated by an element of order $2$ and an element of order $3$. ...