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

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Can this polynomial transformation produce new symmetry?

I've got a polynomial transformation on $\mathbb{R}^6$, and I have a conjecture about it, but I'm having a hard time proving it. The transformation looks like this: $ u:= abcde + abc + abe + ade + ...
2
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1answer
1k views

Distributing $n$ different things among $r$ distinct groups such that all of them must get atleast $1$

In how many ways can we arrange $7$ different things to $3$ people, such that all of them must get at least one? We know that if we have $n$ identical items which will be distributed in $r$ distinct ...
2
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0answers
87 views

Number of permutations possible?

Given two permutation of $1, \ldots, N$. Where 3<=N<=1000 Example For $N=4$ First is $\begin{pmatrix}3& 1& 2& 4\end{pmatrix}$. Second is $\begin{pmatrix}2& 4& 1& ...
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2answers
48 views

Finding a set of a subgroup in Sx

I cannot think of a set $X$ and a subgroup $H$ of $S_x$ which is isomorphic to $\mathbb{Z}$.
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0answers
186 views

Magic Square Combinatorics

This question has been noted to be close to a Project Euler question. Please Help me with this question:Considering a 4*4 magic square ,How many ways are there to fill each square with an integer ...
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2answers
258 views

Probability with combinations of 1-77 numbers and alphabets

(Sorry about me not being able to explain this problem with perfect mathematical terms) Consider the following data set its a combination set of 1-77 ...
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1answer
65 views

Communicating with Language

Can you please help me with this problem? There are $n$ people living on a planet. It is known that their planet has $6$ languages and each person knows every language. It is also known that any two ...
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1answer
46 views

Permutations of sequences.

Find the number of sequences for number of elements = 4. Out of these 4 elements 2 elements are 1 and 3. The remaining 2 elements can have any values between 1 and 3 inclusive. I came up with a ...
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1answer
59 views

Permutation Question of Dihedral Group Order

Let $n \ge 3$. Number the vertices of a regular n-gon by the numbers $1,2,...,n$. Each symmetry of the n-gon corresponds to a permutation of its vertices and hence to an element of $S_n$. The subgroup ...
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1answer
95 views

product of disjoint cycles

Define a permutation $\sigma \in S_7$ by letting $\sigma (1) = 3, \sigma(2)=2, \sigma(3) = 7, \sigma(4)=5, \sigma(6)=1, \sigma (7)=6$. I need help with presenting $\sigma$ as a product of disjoint ...
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1answer
72 views

A subgroup of $S_n$ of odd order is contained in $A_n$

I saw this question. The original questioner asserted that the Lagrange's Theorem is sufficient to solve the problem, but I think that the theorem does just say that the order of $H$ divides the order ...
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3answers
255 views

Decomposition of a cycle as a product of transpositions

Can someone please explain the rules pertaining to different ways to write a cycle decomposition as products of 2-cycles, an example from textbook: I understand this $$ (12345) = (54)(53)(52)(51) ...
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1answer
126 views

Finding the smallest set on which a group acts faithfully

Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the ...
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1answer
91 views

Predicting the Nth combination in sequence of A and B

First I need a way of generating every possible combination of As and Bs in an array of 2048 items. Once I have that would it be possible to row N without action generating every row between 0 and N? ...
2
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1answer
63 views

Signs of products of permutations with given values sums

There is a funny property of permutations, which is valid for $n=2,3,4$, but it would be interesting to know if it is a general fact. Let $\sigma_1,\sigma_2,\sigma_3$ be three permutations of numbers ...
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1answer
80 views

Permutations with repetition for some elements

Suppose we have $N$ slots, each of which can be filled with $X$ options, but $2$ of these slots can only be filled in $1$ way (out of $X$ ways), then what is the number of permutations possible ? For ...
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1answer
122 views

Finding the permutation that shows two permutations are conjugates method?

Problem: Given $\sigma=(12)(34)$ and $\gamma=(56)(13)$ find $\tau\in S_6$ with $\tau^{-1}\sigma\tau=\gamma$ Attempt: I'm kind of new to this but from what I understanding find $\tau$ that satisfies ...
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2answers
49 views

Possible combinations of digits

What is the probability that a random $r$-digit number $(r \geq 3)$ contains at least one $0$, at least one $1$, and at least one $2$? My initial guess was $1-(\frac{7}{10})^r$ seeing that it's $1$ ...
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1answer
65 views

Number of permutation with non-consecutive blocks

How many strings are there consisting of exactly M A's, N B's, and K C's so that the string BC does not appear? For example, when M=3, N=1, K=1, $$ABACA$$ counts as a valid string whereas ...
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1answer
164 views

Number of ways of arranging numbers with given max difference

How many ways are the there to arrange n numbers out of m numbers (1 to m) so that the difference between the max and min numbers of those n numbers is D which is given. For example : n = 4 m = 3 ...
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2answers
365 views

Ways to select 3 members from 5 candidates

At an election there are 5 candidates and 3 members are to be selected. In how many ways a voter can vote? My attempt: 1st member can be chosen in 5 ways, 2nd in 4 and 3rd in 3. So, $5*4*3=60$. ...
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4answers
1k views

Count of 3-digit numbers with at least one digit as 9

Find the number of $3$ digit numbers (repetitions allowed) such that at least one of the digit is $9.​$ I've posted my answer below. If there is a better way to solve this question, I would be ...
0
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1answer
115 views

Every nontrivial subgroup $H$ of $S_9$ containing some odd permutation contains a transposition. [duplicate]

This is a true or false question. Apparently, it is false, but I don't follow. Clearly, if it contains an odd permutation, and an even/odd permutation is defined by the number of transpositions it ...
0
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1answer
34 views

Permutations with some fixed numbers

You have to fill 4 spaces with 3 numbers (4, 5, 6) such that the numbers 4 and 6 appear atleast once in every case. Find the number of such unique permutations. [Ans. 50] How do you go about solving ...
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0answers
45 views

Count of 3-digit numbers

How many different three digit numbers can be formed with the digit $1,2,3,4,5,6,7,8$ none of the digit being repeated in any of the numbers so formed? $120/1200/180/270$ My attempt: $8*7*6=336$, ...
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1answer
68 views

Minimum moves to transform a list to another?

Given two list of n positive elements. We are allowed to perform only one transformation which is to increment each element of the list except one. What are the minimum number of transformation ...
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4answers
480 views

How many ways to arrange Lego bricks on a Lego board?

Let's say I have a board like this one (though significantly smaller, it's 4x7) and I have two 2x3 bricks. I'd like to know how many ways to arrange the bricks on the board. The bricks should ...
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1answer
89 views

Select r items from a set with multiplicity k and total items n.

Let N be a set of n distinct objects having the same multiplicity k. For instance, N={1,1,2,2,3,3} where n=3 and k=2. Now I want to select r numbers from ...
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1answer
252 views

Writing a permutation as products of transpositions

If a can write a permutation $\sigma$ as a product like $\Delta \alpha \beta$, where $\Delta$ is a product of transpositions (in fact, anything) and $\alpha$ and $\beta$ are two disjoint ...
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1answer
2k views

How many distinct 4 letter words can be created using the 26 alphabets.

How many distinct 4 letter words can be created using the 26 alphabets. I have a project in which I can enter the name of the branch ID which is a 4 letter word that can be of any combinations. I want ...
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1answer
55 views

Can an infinite permutatation be decomposed into finite number of infinite cycles?

Let $\sigma \in Perm(\mathbb{N})$ the set of permutations on the naturals. Then can $\sigma$ be written as a finite composition of possibly infinite disjoint cycles?
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1answer
32 views

Restricted permutations revisited!

In how many ways can we arrange $n$ different things at $r$ places (each of $r$ places can have any of the $n$ things)repetition allowed,such that $2$ of the $n$ things are always included? Foe ...
2
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2answers
118 views

$(123)$ and $(132)$ are not in the same conjugacy class in $A_4$

Could you tell me how to show that $(123)$ and $(132)$ are not in the same conjugacy class in $A_4$? I know that all 3-cycles can't be in the same class, because the order of each class must divide ...
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1answer
43 views

Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity when $G$ is infinite?

Let $G$ be an infinite transitive permutation group acting on a set $\Omega$. Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity for $G$ in $\Omega$? $G_\alpha$ is the set of elements of ...
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1answer
81 views

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition? Here $S_9$ denotes the group of all permutations (i.e. bijections with itself) of the ...
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3answers
147 views

Is this possible to convert one array to another given array?

you are given two arrays having n elements , like for n=4,suppose array1={1,2,3,4} array2={2,1,4,5} convert array 1 to array2 performing operation minimum number of time . Also state if ...
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0answers
56 views

Dinner group rotation. Sixteen couples. Four couples per house. Each couple to meet all the others, no repetition.

I want to set up a rotation of sixteen couples with four couples per house so that all couples eventually have dinner together, no repetition. Each couple is to host one dinner. Meetings are monthly ...
2
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1answer
1k views

5 Letter Arrangements of the word 'Statistics'

How many different 5-letter 'words' can be formed from the word 'statistics'? I really am pretty stumped. I understand how to calculate more simpler questions in which each letter of the word is ...
2
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0answers
224 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
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3answers
490 views

In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together?

Problem : In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together? 1st Approach : L T L T L T L T L The 5 lions should be arranged in the ...
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1answer
96 views

The number of permutations in a finite set such that $\sigma(i) \not= i $

Given a finite set $S$ with $|S|= n$, what is the number of permutations $\sigma$ such that $$\sigma(i) \not= i, \forall i \in S $$ That is, permutations where every element is interchanged with ...
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1answer
89 views

Orders of Cycles

Suppose $\tau$ is a cycle of order $n$. I am trying to show that $\tau^k$ is a cycle if and only if $\gcd(n,k)=1$. $\Rightarrow$ If $\gcd(n,k)=1$, then the order of $\tau^k$ is $n/\gcd(n,k)=n/1=n$. ...
0
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1answer
75 views

Combinations and their sum with constraints

I have a number of books (n). They all have different a different thickness and mass. I know that there are (2^n)-1 combinations to place the books. The order of the books does not matter. However ...
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3answers
736 views

How many 90 ball bingo cards are there?

In the UK there are 90 bingo balls. A bingo card consists of 9 columns and 3 rows. A row contains exactly five numbers and four blanks. A column consists of one, two or three numbers and never three ...
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2answers
3k views

A fair die is rolled eight times. What is the probability of getting exactly 2 threes, exactly 3 ones, and exactly 2 sixes?

I was given the following hint: Break the task of counting the desirable outcomes into three subtasks: (a) choose the location of the extra non-three, one, or six in the sequence of eight throws, ...
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2answers
147 views

Size of alternating group $A_n$

This is not too obvious to me - what is the size of alternating group? Following the hint in the comment, should it be $A_n = S_n/2$? So I don't feel right up to here.....
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1answer
156 views

Solving a statistics problem with both permutation and combination

The question I'm having issues with is 17, as shown in the picture. I understand the difference between combinations and permutations, though I'm having trouble applying it to the question. This ...
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2answers
563 views

all possible sequences of positive integers that sum upto N and are strictly increasing

I have $N$ bricks and i have to build a staircase. A staircase will consist of steps of different sizes in decreasing order, no two step size should be same. Each step should consists of atleast one ...
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0answers
60 views

Count possible decodings for given number

If A = 1, B = 2, C = 3,....,Z = 26 How to count possible decoding for given any integer number? EXAMPLE : NUMBER : 111 --> ANSWER : 3 EXPLANATION : ...
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2answers
141 views

Possible ranks of a matrix

Let $v=(a_1,\cdots,a_n)$ be a real row vector. We may form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the entiers of $v$ in all possible ways. The rows can be listed in an ...