Questions tagged [permutations]

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

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Six-letter ‘words’ are formed using the letters A, B, C and D. In how many of them does each letter appear at least once?

Six-letter ‘words’ are formed using the letters A, B, C and D. In how many of them does each letter appear at least once? In the answer, it says $$ 4 \cdot 6 \cdot 5 \cdot 4+\left(\begin{array}{l} 4 ...
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2 votes
1 answer
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Proving there aren't any more subgroups for $S_3$

I'm currently doing an exercise to find all the subgroups of $S_3$, with a hint given that there are exactly $6$ and then to prove that no more subgroups exist. Take $$() \equiv e, (12) \equiv x, (13) ...
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-5 votes
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How many different 5-letter strings are possible? [closed]

I'm assuming a 26-letter alphabet with 5 letters picked randomly. Repeats are allowed and order matters. How many possible permutations (if that's the right word)?
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How many permutations are there of the letters of the word AARDVARK? In how many of the permutations are the A's separated?

How many permutations are there of the letters of the word AARDVARK? In how many of the permutations are the A's separated? I got answer for the first part of the question, but for the second part I ...
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Sum of product of characters from slightly different permutation groups

Let $S_n$ be the group of permutations acting on the set $\{1,...,n\}$. Given $R_1,R_2$ two irreps of $S_n$ with characters $\chi_{R_i}$, I know that $$ \sum_{a\in S_n}\chi_{R_1}(a)\chi_{R_2}(ab)=\...
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How many permutations are there for PARALLELOGRAM where the A's are separated?

In my book there is a permutation example question, the question is: How many permutations are there of the letters of the word PARALLELOGRAM? In how many of these are the A's separated? I ...
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1 vote
3 answers
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How to use exponential generating functions to count the number of k-letter permutations from n letters?

I was learning about exponential generating functions and stumbled upon the following question and answer : Sample question Given the string of letters ABBBBBBBBBBBBBBBBBCDEFGHIJKLMOPQRSTUVWXYZ (that'...
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1 answer
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The commutativity of $(12)(34)$ and $(12)$ sufficient to say that : the $S_5$-conjugacy class of $(12)(34)$ is also an $A_5$-conjugacy class?

I read in a document about the symmetric group. I came across a paragraph that I didn't understand why, it's the following: One checks that $(123)$ commutes with the odd permutation $(45)$. Therefore,...
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-3 votes
3 answers
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Possible 4-digit numbers using ${1,2,3,4,5}$ under constraints.

How many 4-digit numbers can be crafted from $\{ 1,2,3,4,5 \}$ under the following conditions: $1$ can not appear two or more times ($1142$) is not valid $2$ can not appear three or more times ($2242$...
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Signature of a permutation couple

I was working on a problem when I found out I needed to know the signature of a permutation of the form : \begin{equation} (\sigma_A,\sigma_B) \end{equation} meaning that $(\sigma_A,\sigma_B)$ is ...
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1 answer
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Different Division Combinations In NFL

I am running an optimization to try to determine the best division alignment in the NFL. There are 32 teams in a league, in which 8 divisions are made up of 4 teams each. There are 35,960 potential ...
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1 vote
1 answer
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Elementary Probability: What is the probability of picking a permutation of the letters NUMBER which starts and ends with a vowel?

What is the probability of picking a permutation of the letters NUMBER which starts and ends with a vowel? I thought that the there are two situations: when the u is in front and when the e is in ...
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Calculating a maximum size of subset of permutations and finding an example of such subset. [closed]

I've been trying to solve a problem of finding a subset of permutations under a certain constraint. So far I wasn't able to solve this, hope someone can help. Thank you in advance. The problem: We ...
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How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child?

Problem How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child? The seats are not numbered. My Opinion I ...
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I am trying to generate random matrix with based on some condition, How many matrices the can be generated?

I am trying to generate random matrices based on the following conditions. There will be 3X3 matrices. The first column can have 1 to 10, the second column can have 11 to 20 and the third column can ...
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Prove that factoradic method gives $k$-th lexicographical permutation.

Factoradic method can be used to get $k$-th lexicographic permutation of $n$-elements. The factoradic method is described here: https://en.wikipedia.org/wiki/Factorial_number_system#Permutations. ...
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1 vote
0 answers
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Is there a efficient way to find only the combinations of interest from V exciting in P [closed]

Is there a efficient way to find only the combinations of interest from V exciting in P I have a list of something like this. V = [2,3,5,7,9,12] Size of V can be ...
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-2 votes
0 answers
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Group rotations help please! [closed]

We have 16 groups (school families) we pair them up for field day for a total of 8 groups of 2 families each. I need to make stations in which there are no repeats in the stations at the complete and ...
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-4 votes
1 answer
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Is the problem below a Permutation or Combination? [closed]

In how many ways can a boy change his clothes 5 times from 5 pants and 7 polo if he must need to attend 3 parties?
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1 vote
2 answers
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How many different $2 \times 2$ Sudokus are there?

PROBLEM How many different $2 \times 2$ Sudokus are there? APPROACH This seems pretty easy to brute force. There are $576$ Latin squares of size $4$ (which are the sudokus without restriction on boxes)...
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Sending messages to Aliens with given restrictions using Combinatorical Technique

I made up this question ,however i stuck in it ! Assume that we get in contact with primitive alien clan , and they have very strange language such that Their alphabet only consist of the ...
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3 votes
1 answer
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Possible combinations of letters

I have an example in my lecture where I have the letters: A,A,A,B,B,C,C,C and I need to give the possible combinations. $$ \frac{8!}{3!\cdot3!\cdot2!} = 560 $$ In one part of the exercise I have to ...
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Order of Affine non-soluble groups

2-Transitive groups has been classified. The complete table has been mentioned in this Textbook ( see e.g., Table 7.3 and Table 7.4, page no. 194-197). Table 7.3 contains Affine 2-transitive groups ...
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Calculating Expected Value from Permutation

Two 2-digit numbers are formed by randomly selecting digits, without replacement, from the digits 1,2,...,9. What is the expected value of the product of the two numbers? I figured that there are 72 ...
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1 vote
1 answer
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Index of Socle of primitive permutation group $G$

This is a follow up question of this (Edit: The following question is for primitive group $G$ which lies in the case (i) and does not lie in case (ii) of Theorem 5.6C of the textbook "Finite ...
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1 answer
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Multiplication of permutation Cycles

I was doing example 1 Page 94 from Joseph A Gallian , In question it was given $\alpha = (231)$ and $\beta = (132) $ and then on next page there is $$ \alpha\beta = (21) $$ . So if we think ...
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A permutation problem regarding number of ways of a given permutation

Given a permutation of N length. Lets say the permutation is : p1,p2,....,pn. How many tuples [a,b,c,d] such that: pa < pc and pb > pd. Example: 5 3 6 1 4 2 for this permutation of 6 length, ...
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3 votes
1 answer
138 views

number of permutations maximizing a sum

Let $n$ be an odd integer greater than $1$. Find the number of permutations $\sigma$ of the set $\{1,\cdots, n\}$ for which $|\sigma(1) - 1| + |\sigma(2) - 2|+\cdots + |\sigma(n) - n| = \frac{n^2 - 1}...
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Permutations of $[n]$ where no 3 numbers congruent $\mod k$ are consecutive

Question: Given natural numbers $k$ and $n$, in how many ways can we order the numbers $1,2,\dots, n$ such that no three consecutive numbers are congruent $\mod k$? I was reading this paper where ...
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Permutation test statistic is invariant under permutations

I have built the asymptotic test with statistic $T_n = T_n(X_{11},\dots,X_{1n_1}, X_{21}, \dots, X_{2n_2} )$ (the sample is obtained from one with sample size $n = n_1 + n_2$ by dividing into two ones ...
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3 votes
1 answer
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Is the Socle of an almost simple group a simple group?

Let $G$ be a finite primitive group of degree $n$, and let $H$ be the socle of $G$. Then if $H$ is isomorphic to a direct power $T^m$ of a nonabelian simple group $T$ then the following holds when $m=...
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1 vote
0 answers
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What transitive action of a permutation group connects the Prufer 2-group?

Let $X$ be the dyadic rationals in the half-open unit interval. The graph $G$ over $X$ having the vertices $(x,x+2^{\nu_2(x)-1})$ and $(x,x-2^{\nu_2(x)-1})$ is connected. It's just the infinite ...
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10 votes
7 answers
1k views

How to find the password space given several restrictions?

I am trying to determine all valid passwords (the password space) that fulfill this list of requirements. Password is exactly 15 characters long. Must contain at least 2 lowercase letters (26 in ...
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1 answer
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Combinatorics related math problem...need suggestion

I am trying to solve the following question: "In a programming class of 7 students, the instructor wants each student to modify the program from a previous assignment; however, no student should ...
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4 votes
2 answers
134 views

Question from isi previous years

(a) Show that $\left(\begin{array}{l}n \\ k\end{array}\right)=\sum_{m=k}^{n}\left(\begin{array}{c}m-1 \\ k-1\end{array}\right)$. (b) Prove that $$ \left(\begin{array}{l} n \\ 1 \end{array}\right)-\...
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0 answers
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How do I represent hexagons (or any polygon really) so that they are equal independent of rotation?

I am working on a private coding project. My program works with hexagons, which edges have certain features. I currently represent these hexagons using arrays with one entry for each edge. There is a ...
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The difference between permutation and composition in the subject of probability

In general we have: $$\dfrac{p(k,r)}{p(n,r)}=\dfrac{c(k,r)}{c(n,r)}$$ is it different or same in the topic of probability to solve the question by permutation or combination? For example, from 5 green ...
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-1 votes
2 answers
56 views

how many three letter permutations are there of the letters of the word EQUILATERAL [closed]

I tried 11P3/(2!x2!x2!) and then I tried (11!/(2!x2!x2!))/8! but always end up with a decimal number and I don't understand why this won't work.
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3 votes
0 answers
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Is a transitive group more likely to be soluble if its degree has more prime divisors?

There is some empirical evidence that the probability that a permutation group of a given degree is soluble increases with the number of prime divisors of the degree. (Primes are counted with ...
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2 votes
2 answers
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the number of ways to order a line from a group of 6 women and 2 men such that 2 men stand on either end?

Given a group of 6 women and 2 men, I have to find the number of ways in which atleast one man is on one end of the line. I tried to separate the cases where both men take the ends and when only one ...
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0 votes
1 answer
57 views

Why can we not use combinations?

. A 3-person basketball team consists of a guard, a forward, and a centre. a) If a person is chosen at random from each of three different such teams, what is the probability of selecting a complete ...
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1 vote
1 answer
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Order of $U_{27}$, 2 Answers?

In my book I saw: $$U_{27} = \{1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 26\}.$$ I know that order of group is the number of elements inside that group, so we get an order of $...
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0 votes
0 answers
37 views

Trying to relate information theory basics and physics

I am a layman interested in information, and wondered how far off this idea is: Imagine 2 levels and 3 particles (in the physics sense, indistinguishable), and the different permutations: 1 arrange $$...
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  • 13
-1 votes
4 answers
91 views

Count of 3 digit numbers having digits in non-increasing order [closed]

What is the count of 3-digit positive numbers such that all the digits(from left to right) are in non-increasing order of value? For e.g: 633 is counted, as its digits from left to right are in non-...
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1 vote
0 answers
48 views

Number of unique sequences of length 7, where elements are members of the digits 1 through 5, and repeated elements are allowed.

I need a little help with this practice problem of mine. I believe I am a little stuck on what it means by "unique" sequences. I know that if such a word wasn't there. The answer would ...
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-2 votes
1 answer
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If $G$ is a transitive permutation group of even degree $>2$, then 4 divides $|G|$.

I am struggling with the following problem: If $G$ is a transitive permutation group of even degree $>2$, then 4 divides $|G|$. What I mean by $G$ being a transitive permutation group of even ...
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2 votes
1 answer
154 views

An efficient algorithm to generate a set of tuples satisfying a given upper bound for a distance between two arbitrary elements

Let $T_i^n$ denote a particular tuple of $n$ natural numbers (here $i < n!$ and we assume that the tuple contains all elements of the set $\{0, 1, \ldots, n-2, n-1\}$, i.e. there are no duplicates)....
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4 votes
1 answer
84 views

Doubt with Socle and O'Nan-Scott Theorem.

The following is the statement of O'Nan-Scott Theorem. Theorem: Let $G$ be a finite primitive group of degree $n$, and let $H$ be the socle of G. Then either (a) $H$ is a regular elementary abelian $p$...
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0 votes
1 answer
83 views

Order of largest subgroup of $S_n$ [closed]

I know that $S_{n-1}$ is a maximal subgroup of $S_n$, but is it also maximum? I.e., what's the size of the largest subgroup of $S_n$? Is it $(n-1)!$
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  • 27
2 votes
0 answers
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$\sigma \in S_n$ does not commute with any odd permutation if and only if the cycle type of a consists of distinct odd integers.

This is (part 1 of) Exercise 4.3.21 in D&F 3ed Abstract Algebra, which I have to prove: Show that $\sigma \in S_n$ does not commute with any odd permutation if and only if the cycle type of $\...
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