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

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What's an effective way of comparing orderings?

Given two orderings of some set of things, what's an effective way to quantify how similar the orderings are? For instance, say I have the orderings ...
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1answer
76 views

Permutation of natural numbers [closed]

Find the number of permutation of {1,2,3,4,5,6} such that the patterns 13 and 246 do not appear. Show the steps .
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1answer
27 views

Help with Discrete Probability

I'm trying to show statistics related to combinations of discrete variables. I have a feeling what I need to do is quite simple but I can't for the life of me remember the correct formulas to achieve ...
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1answer
41 views

Question about cyclic (Renew Question) [duplicate]

At my Question about permutation - (I add new details, now it shold be more clear)(Question about permutation - (I add new details, now it shold be more clear) I explain it wrong , now I try to ...
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1answer
150 views

Finding the “square root” of a permutation

Suppose $r$ is odd, and ${\rm ord}(\alpha)=r$. ($\alpha,\beta$ are cycles.) Now, $\alpha=(a_1\cdots a_n)$. I need to find $\beta$ that will make $$\alpha = \beta^2$$ How can I show it? What I ...
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1answer
151 views

Determining the Probability of a Password

I create a 10 letter password by randomly mixing: 4 letters, from (A, B, C, D, or E) without repeats 3 numbers, from (1, 2, 3, 4, or 5) without repeats 3 symbols, from (@, #, $, %, or *) without ...
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1answer
81 views

How to show that two groups makes $S_n$

I need to show that: $S=\left\{(12),(13),...,(1n)\right\}$ generates $S_n$ $S=\left\{(12),(123\cdots n)\right\}$ generates $S_n$ How do I show that each one of them generates $S_n$? Thank you!
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1answer
106 views

Permutation and combination of letters

I need help with the following question: "Given that a computer can only type letters A,B,C,D and E, how many ways can I type in 6 letters such that they must contain at least all of the different ...
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4answers
419 views

Number of $2n$-letter words using double $n$-letter alphabet, without consecutive identical letters

How many words with $2n$ letters can be created if I have an alphabet with $n$ letters and each of the letters have to occur exactly twice in the word, but no two consecutive letters are equal? ...
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1answer
113 views

Question about permutation cycles

We have: $\alpha = (a_1a_2 \cdots a_r), \beta=(b_1b_2\cdots b_r)\in S_n$ ($\alpha,\beta$ are strange cycles) How can we find $f\in S_n$ s.t.: $$\beta=f\alpha f^{-1}\;\;\;?$$ Thank you! (The answer ...
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2answers
33 views

Number of permutation

I can't guess how to compute number of permutations when I have $f$ fields and number $a$ of elements to distribute among them. Such that sum must be equal to $a$. Could you enlighten me and refer to ...
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2answers
91 views

How to multiply permutations

I didn't understand the rules of multiplying permutations. I'll be glad if you can explain me... For example: we have $f=(135)(27),g=(1254)(68)$. How do I calculate $f\cdot g$?? Thank you!
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2answers
206 views

probability and combinations with the word REGULATIONS

If the letters of the word REGULATIONS are arranged at random,what is the probability that there will be exactly 4 letters between R and E? The answer in my book is given as 11!/(9C4 x 4! x6!x2!) ...
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1answer
87 views

How many permutations of $\{1,2,3,4,5\}$ leave at least two elements fixed?

How many permutations $f: \{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ have the property that $f(i)=i$ for at least two values of $i$? I'm just struggling with this inclusion/exclusion question. I ...
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1answer
220 views

Raising a cycle to a power, cycle decomposition

Let $\alpha$ be an m-cycle. Is it true that $\alpha ^k$ can be decomposed into $\gcd(m,k)$ disjoint cycles? for example $(1 2 3 4 5 6)^2 = (1 3 5)(2 4 6)$, $(1 2 3 4 5 6 7 8)^6 =(1 7 5 3)(2 8 6 ...
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2answers
199 views

Is the converse of Lagrange's Theorem true for the permutation group $S_5$?

Is the converse of Lagrange's Theorem true for the permutation group $S_5$? That is, if $n\mid |S_5|$, then is there a subgroup of $S_5$ with order $n$. Since $|S_5|$ = 5! = 120, then any subgroup ...
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2answers
166 views

Prove that $PSL(2,9)\cong A_6.$

I have tried to solve the Exercise 8.12, page 227, in Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate texts in Mathematics, Springer-Verlag, New York (1995). The ...
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1answer
157 views

number of derangements

In the normal derangement problem we have to count the number of derangement when each counter has just one correct house,what if some counters have shared houses. A derangement of n numbers is a ...
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1answer
83 views

Permutations of letters under some conditions

Let $W(p,q,r,s)$ be the number of permutations of the letters which satisfy the following conditions : Condition 1 : The letters are consist of $P,Q,R,S$. Condition 2 : The number of letter ...
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1answer
44 views

Prove whether or not $H$ is a subgroup of $S_n$

$H$ is the set of permutations where $H$ = {$ID_{S_n}$,(12),(34),(12)(34),(13)(24),(14)(23),(1432),(1234)}. Is $H$ a subgroup of $S_4$? Is there a simpler way to do this than checking for ...
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370 views

A croissant shop has plain, cherry, chocolate, almond, apple, and broccoli croissants.

a) How many ways are there to choose a dozen croissants? b) How many ways are there to choose three dozen croissants? c) How many ways are there to choose two dozen croissants with at least two of ...
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3answers
93 views

How many possibilities in tinyurl

Looking at tinyurl, there is anywhere from 1 digit to 7 digits of I believe 36 choices (lowercase letters a to z and digits 0 to 9) How do I calculate mathmatically the number of permutations of the ...
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1answer
115 views

Permutation of Combination. How do I work out this example?

I have this question which is raising how I can differentiate permutation from combination. I have the word BOOMBASTIC. a) Find the number of different selections of 3 letters that can be formed ...
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1answer
143 views

If a subgroup of a symmetric group has an odd permutation, then it has a subgroup of index 2.

I want to show that for $n\geq 2$ and $H\leq S_n$ if $H$ contains an odd permutation then it necessarily has a subgroup of index 2. I am not sure how to start, if it has an odd permutation then it ...
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1answer
140 views

Combinatorics and Derangements

Determine the number of permutations of $\{1, 2, \dots, n\}$ in which no odd integer is in its natural position. I'm having a hard time generalizing this situation for $\{1, 2, \dots, n\}$ When ...
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1answer
82 views

How to show that an automorphism of $S_n$ is inner?

Given an automorphism $\phi:S_n\rightarrow S_n$ such that it maps all the transpositions on the transpositions, how do I show that this map is given by a conjugation with an element $s\in S_n$? ...
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2answers
402 views

Permutations and Derangements

Determine the number of permutations of $\{1,2,...,9\}$ in which at least one odd integer is in its natural position. __ __ __ __ __ __ __ __ __ There are nine numbers to permute in the $9$ ...
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1answer
55 views

permuting digits of a number

Given a number $N$ with upto 18 digits, We need to find how many numbers smaller than $N$ can be formed using the same digits. Eg: for 725 we can form 527,572,257,275 hence answer=4
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1answer
41 views

transformation of DFT matrix

$\mathbf{F}$ is a unitary DFT matrix where the $(m,n)$-th entry of $\mathbf{F}$ is given by $\frac{1}{\sqrt{M}}e^{-\imath2\pi(m-1)(n-1)/M}$. Note that $\imath=\sqrt{-1}$. Let $\mathbf{A}$ be a matrix ...
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1answer
60 views

Combination and Permutation- examples

I am not sure how to answer this question. I have a word TOKYO. a) how many different can I arrange the letters in a row? b) in how many of these will O's be together? c) how many will the two O's ...
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1answer
238 views

how many types of calendars should he prepare

Could anyone help me how to solve this one?. Thank you for explantion. A mint preparess metallic calendars specifying months, dates, and days in the form of monthly sheets lone plate for each month. ...
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1answer
109 views

How does Knuth's second algorithm to calculate permutations work?

I have started reading the Art of Computer Programming Volume 1 by Knuth. The first half of the book is basic concepts in maths. On page 45 there is an algorithm to obtain the next (amount of) ...
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1answer
50 views

The name of certain permutations.

The permutations I'm looking at are 2341, 2413, 3412, 3421, 4123 and 4312. I'll explain the property with the example 2413: I start with the first digit (2) and go to the position 2. There I see the ...
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1answer
77 views

Circular permutations - confusion in textbook example

I have a confusion in one of the examples given in the book Discrete Mathematics: Elementary and Beyond. The exercise says: In how many ways, in a party of six guests, can the people be seated ...
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1answer
54 views

Intersection of Two Permutations

If you are generating 2 digits codes $[X, X]$ where you have 5 choices for $X$. The number of choices for each code is: Order matters so the total number of permutations is $n^r = 5^2 = 25$. If for ...
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2answers
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Prove that $S_n$ satisfies the following property: if $g \in S_n$, then $g$ and $g^{-1}$ are conjugate in $S_n$.

So I tried setting up an arbitrary $g$ such that $g(a_1)=b_1, ... g(a_k)=b_k$, and then I fizzled out. So I'm showing that for every a in $S_n,\ ag(a^{-1}) = g^{-1}$. I'm showing that they have the ...
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2answers
518 views

When to use Permutations or Combinations

Suppose I have a bundle of crayons, I have 5 different colours of crayon (Blue, Black, Brown, Red, Grey) how many unique bags can I create with 10 items per bag such that each bag has at least one ...
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1answer
216 views

Dealing a 5 card hand with exactly 1 pair

Here's the question: A 5-card hand is selected from a standard deck of playing cares. (A standard deck has 13 cards from each of 4 suits{clubs, diamonds, hearts, and spades. The 13 cards have ...
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1answer
363 views

Probability of the sum of two drawn cards being less than 11?

An Ace has a value of 11 in this problem, and a face card has a value of 10. Thus, my idea for solving this problem was to separate the problem into 7 different cases (drawing a 2 first, 3 first, 4 ...
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2answers
98 views

Number of subgroups of $S_4$

How to find the number of subgroups of $S_4$ and list them? For order $1$ it is {${e}$}, order $2$, we will consider $<(12)>$, $<(23)>$, $<(34)>$, $<(13)>$, ...
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1answer
44 views

seating correctly

There are 6 girls who are assigned 6 places to sit in a classroom. In how many ways can they be seated so at most 1 girl is in the correct place? How do you even start with this? My teacher didn't ...
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0answers
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Exceptional isomorphisms of classical algebraic groups

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. An affine algebraic group $G$ is an affine variety over $k$ with a group structure such that multiplication and inversion are ...
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2answers
554 views

Prove that in $S_n$ there are an equal number of even and odd permutations.

Prove that in $S_n$ there are an equal number of even and odd permutations. $S_n$ is a group of all possible permutations on a set of $n$ elements. For this problem we can assume $n>1$. I'm ...
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638 views

How many permutations (bijections) are there on the set B = {0,1}^(8) of bytes? How can there be permutations if there is no function?

I know that B would be the set {00000000, 00000001, ..., 11111111}, and there are 256 elements on this set. I don't know how there can be a bijection on the set, though...I thought that you needed a ...
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What is the total number of permutations of n digits where all digits will exchange their positions? [duplicate]

As we know, Total number of permutations of the digits 12345 is 5!. Well, I am looking for another interesting fact. How many permutations among them are in the way that no digit is in its original ...
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3answers
275 views

Why factorials when divided by factorials less than the number have a remainder 0?

Lets take the example, if we take the expression $\frac{X!}{y_1!\cdot y2!\cdots y_n!} $as long as Summation $S=y_1+y_2+...y_n$ is less than or equals $X$, the remainder is always $0$. Thats How the ...
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1answer
330 views

How many unique arrangements of the word REVERSE are there with the V and S separated?

Calculating the number of unique arrangements of the word REVERSE, taking into account the repeated Es and Rs, is straightforward: $$\frac{7!}{2!\cdot3!}=420$$ I am not sure, however, to calculate ...
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1answer
53 views

Solving a permutation/Combination equation

please help check if this would the correct way to solve this: $^nP_2 = ^{n+1}C_3$. I want to solve for $n$. theoretically, I was thinking that: $^nP_k = k!\times ^nC_k $ hence: ...
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2answers
323 views

How Can I calculate number of combinations/permutations with certain rules

Lets say I have 4 balls and when each ball is drawn it can be any value between 1-40 inclusive. If order isn't important then it would just be $40\cdot 39\cdot 38\cdot 37/4!$ But what if ball 1 had ...
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1answer
70 views

Restoring Permutations

I was curious if anyone knew of any proofs or knew of how one might go about proving problems involving restoring permutations. An example of the type of proof I am interested in is: Prove that ...