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

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0answers
143 views

How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
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2answers
567 views

What will be total number of solutions of $a+b+c = n$?

Please tell me how to find the total number of intergral solutions of $$ a+b+c=n $$ I already know that total number of solutions will be $(n+3-1)c(3-1)$. But what will be the case when a varies from ...
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0answers
66 views

$i$ balls to paint $k$ colors, and exactly $k' < k$ colors should be used. How many ways to paint?

Another ball-painting problem: assume that we have $i$ balls (with numbered labels, so order is sensitive), and $k$ different colors. Now we need to paint these balls using these colors so that ...
1
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1answer
238 views

The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
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2answers
77 views

Can two finite sequences be considered permutations if their products and sums are equal?

Given two finite ordered sequences with possibly non-unique elements all greater than one: $A,B \in \mathcal{Z}_{>1}$. Given that we have: \begin{eqnarray} |A| &= |B| \\ \Pi_{x \in ...
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1answer
72 views

Cycle types of $S_4$

I understand the possible cycle types of $S_4$ are $(4), (3,1), (2,2), (2,1,1), (1,1,1,1)$, but why are there $6, 8, 3, 6, 1$ of each respectively? Also why do the elements of each cycle type form a ...
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2answers
159 views

Problems about symmetric groups

We just finished a lesson about determinants and the symmetric group with all what comes with it ( permutations, transpositions etc... ), except we didn't do group theory ( we only see it next year ), ...
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1answer
156 views

permutation of “counting out”

Josephus problem*: circle=1,2,3,4,5,6,7,8,9,10. count=2. (Beginning at 1) The "last man standing" in this case=9. Order of elimination or permutation (?): 2,4,6,8,10,3,7,1,9 For any size circle and ...
2
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1answer
243 views

Understanding fundamental principles of counting.

There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication. I often got confused applying them. I know that if there are two ...
2
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2answers
146 views

15Puzzle, sum of inversions - what's been summed?

According to this page: http://mathworld.wolfram.com/15Puzzle.html it says that While odd permutations of the puzzle are impossible to solve I've red this article: ...
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1answer
81 views

What is the number of functions $f : A\rightarrow A, \forall_{x\in{A}} f(f(x))=x$, set $A$ have $n$ distinct elements.

What is the number of functions $f : A\rightarrow A, \forall_{x\in{A}} f(f(x))=x$, set $A$ have $n$ distinct elements. For $A=\{a,b,c,d\}$ we could look on the right-upper triangle of matrix of all ...
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1answer
226 views

Adjacent transposition - explain in layman terms

I've red this article many-many times, as well as some more supplementary reading on groups, permutations, generating sets etc. and I give up. :( I'm not a mathematician (I study combinatorics, but ...
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2answers
149 views

Probability of random shuffling of cards

I have a pack of cards and use the following method to shuffle them Pick a random card from the deck and replace the first card with it Put the first card back in the deck Move to the second card ...
2
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0answers
31 views

order of elements in a partition using Maple

I determined this whole partition but I just want to have the finer the partition for example: I have this ...
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1answer
607 views

Conjugacy classes in $A_n$.

Suppose $n$ is a non negative integer $\geq 4$ and $\sigma\in S_n$ a permutation. Conjugacy classes in $S_n$ are completley determined by the cycle structure of $\sigma$. If we let the alternating ...
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2answers
313 views

permutations of a multiset having symbols with fixed multiplicity

Let $N$ be a multiset of $n$ distinct objects having the same multiplicity $k$. For instance, $N=\{a,\,a,\,b,\,b\}$ where $n=2$ and $k=2$. I was looking for the problem of counting the number of ...
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1answer
150 views

Find the subgroups of index two of this finite semi-direct product

This older stackoverflow question may be helpful in answering the question that I ask below, although I could not work it out. For $n\geq 1$, let $X=\lbrace 1,2, \ldots ,n \rbrace$, $Y=X \cup (-X)$ ...
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5answers
234 views

Bijection proof

I got this on an exam and struggled to complete it, could anyone offer a proof? Thanks! Let $X$ be a finite set. Let $f: X \longrightarrow X$ be a bijection. For $n \in \mathbb{Z}^+$, set $$f^n ...
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2answers
156 views

compositions of permutations

For compositions of permutations on a set $X = \{1,2,3\}$, my lecture notes say that the composition $\phi_2 \phi_1$ is the permutation $\phi_1$ followed by the permutation $\phi_2$. So consider the ...
3
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1answer
308 views

Sliding blocks puzzle

Consider a 'game' played on a subset $S$ of an $n^2$ square grid as follows. There are 3 types of pieces, each occupying a square, 1 green, some red and the rest are blue, a move consists of shuffling ...
2
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1answer
46 views

A statement about $S_n$: pigeonhole?

Let $n \in \mathbb N$, $n \ge 2$ and let $S_n$ be the symmetric group on $n$ elements. I will call for shortness $I_n := \{1 , \ldots , n\} \subset \mathbb {N}$. Fix $i_0 \in I_n$ and consider the ...
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0answers
100 views

Random Permutation Poisson proof

Let $F$ be the number of fixed points of a random permutation on $n$ items. Show that as $n$ approaches infinity, the distribution of $F$ approaches a Poisson distribution with a mean $(\lambda)=1$.
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1answer
421 views

Random permutation problem

Let $\pi$ be a random permutation of $n$ objects and let $ T := \text{the number of transpositions in } \pi $. Use Chebychev's Inequality to find an upper bound for $T\geqslant k$. Okay the problem ...
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1answer
131 views

Permutation & Combination wordproblems

I'm studying Permutation & Combination those days and I've got well understanding the whole chapter but those word-problems related to it can't got them well, not even understand any of them. for ...
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1answer
2k views

Number of ways of choosing $m$ objects with replacement from $n$ objects

There is a set of $n$ distinct objects. How many possible multisets can we get when choosing $m$ objects with replacement? Note that the elements in a set are unordered and distinct, and the elements ...
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1answer
149 views

what is the formula to calculate the permutations

I am new to the permutations. I have a problem with me for which I am not able to use proper formula - Problem: There are X boxes in which balls need to be placed. The balls are of two colors - BLUE ...
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1answer
92 views

Permutation for balls [duplicate]

Possible Duplicate: number of combination in which no two red balls are adjacent. We have $N$ slots. They have to be filled with balls (either green or red), one ball for each slot. ...
2
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1answer
293 views

Why does multinomial theorem only works for identical set of objects?

I will elaborate this with an analogy, 15 toys are to be distributed amongst 3 children , such that any child can get any number of toys, so we have to find the number of ways in which we can do so ...
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3answers
737 views

Number of four digit numbers?

I came across this question, that how many 4 digit numbers $a_1a_2a_3a_4$ are there such that $a_1\geq a_2\geq a_3\geq a_4$ I know the answer for this case, $a_1>a_2>a_3>a_4$ I ...
7
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1answer
602 views

number of combination in which no two red balls are adjacent.

given x spaces(you can fit 1 ball in 1 space) and unlimited number of identical red and white balls, find the total number of combinations in which no two red balls are adjacent to each other. i ...
2
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2answers
684 views

Irreducibility of the standard representation of $S_n$.

The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
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1answer
707 views

How many combination of numbers

Anyone can give me an idea how I approach calculating the following problem. How many possible valid numbers, where a valid number is any number between 0-9, length of 10 digits, excluding # or *, a ...
2
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2answers
616 views

How does one decompose the regular representation of $S_3$?

I need to decompose the regular representation of $S_3$ into irreducible ones. What I know so far is this: $S_3$ is generated by $\tau = (12)$ and $\sigma = (123)$. If $v$ is an eigenvector of ...
2
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1answer
1k views

20 books 5 different shelves

So I'm trying to answer this question and am not sure if my answer is correct. In the text book I'm using, this question asked before combinations are even introduced (only permutations) so I'm not ...
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1answer
167 views

How many possible ways are there

Suppose i have the given data set of length 11 of scores p=[2, 5, 1 ,2 ,4 ,1 ,6, 5, 2, 2, 1] I want to select 6 ,5 , 5 , 4 , 2 , 2 scores from the data set. How many ways are there? For the above ...
2
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1answer
346 views

Simple Dice Rolling Problem

If you play poker dice by simultaneously rolling 5 dice, why is $P\text{{five alike}} =.0008$? I guess I understand the fact that each dice has the probability to land on the same number $1/6$ of the ...
2
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2answers
390 views

$n$-permutations with exactly $k$ fixed points

It's easy to deduce the formula for $n$-permutations with exactly $k$ fixed points. The result is similar to $n$-derangement formula and it's equal to $ D_{n,k}= ...
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1answer
111 views

Find The number of ways?

The number of ways in which all the integers from 1 to 36 (both inclusive) can be arranged such that no two multiples of 6 are adjacent is expressed as $$ m! x^n Pr $$ where m, n, r are distinct ...
0
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2answers
97 views

Permutations question

I was doing this question: Repetitions not allowed: Using the following six digits: 2, 3, 5, 6, 7, 9 What is the probability that a three-digit number greater than 400 will be formed from these ...
1
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1answer
129 views

Permutation of a group

If we have the following permutation: $\sigma = (2,1,4)(4,5,1,6) \in S_7$ Am I right in assuming it will permute in the following way? $ \begin{array}{clcr} 1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \\ 6 \ 1 \ 3 \ ...
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1answer
75 views

Derived Subgroups G'

Can someone tell me if the following proposition is true ? let $G$ be group and $g_i\in G$ for $i=1\ldots n$ for any permutation $\sigma\in S_n$ then we have $g_1g_2\ldots g_ng^{-1}_{\sigma(1)}\ldots ...
2
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1answer
398 views

Combinatoric puzzle

During a dinner with $k=20$ persons sitting at $n=4$ tables with $m=5$ seats, everyone wants to share a table with everyone. The assembly decides to switch seats after each serving towards this goal. ...
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2answers
176 views

Find number of possible arrangements of N disks with given constraint

In how many ways you can make a stack of N disks, such that: Bottom disk always has radius 1 A disk can be placed on the stack if it radius is <= (maximum of all disk radii below it + 1) You ...
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3answers
711 views

What is the correct terminology for Permutation & Combination formulae that allow repeating elements.

Let me explain by example. Q: Given four possible values, {1,2,3,4}, how many 2 value permutations are there ? A: 16. ...
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2answers
54 views

Jacobi symbol and invertibility of $m$ for an odd $n$

I have asked a similar question here before, and received a nice answer. I think that the next question here is equivalent, but can't seem to be able to prove it. Here goes: Given an odd $n$, I want ...
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0answers
108 views

How many ways to fill the $N \times N$ board by nonnegative integers, such that sum of the numbers of each row and each column is $R$?

How many ways to fill the $4 \times 4$ board by nonnegative integers, such that sum of the numbers of each row and each column is $3$? I wrote a brute-force and got $2008$ which seems to be the ...
0
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2answers
190 views

Determine the numbers of permutations $\sigma$ so $\gamma '= \sigma \gamma \sigma ^{-1}$

I have a question regarding permutations. If $\gamma = (123) (45) (6)$ and $\gamma ' = (1)(23) (456)$ in $S_{6}$ how do I then determine the numbers of permutations $\sigma$ in $S_{6}$ so ...
3
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1answer
570 views

Permutation questions

What is the smallest number n such that $A_{n}$ contains a permutation of order 2004? I calculated it to 334, but the answer is 176 and I can't see why? I first noticed that $2004 = 2^{2}\cdot 3 ...
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1answer
59 views

Permutation of order 12 and 30 in $S_{9}$

If I have the group $S_{9}$ and $\sigma, \tau \in S_{9}$ where $\vert \sigma \vert = 5$ and $\vert \tau \vert = 6$ is it then possible to have $\vert \sigma \tau \vert = 30$ and $\vert \sigma \tau ...
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2answers
525 views

How many seven-digit numbers satisfy the following conditions?

How many seven-digit numbers divisible by 11 have the sum of their digits equal to 59? I am able to get the seven-digit numbers divisible by 11 and I am also able to get the seven-digit numbers ...