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

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Examples of injective maps such as MTF (Book Stack). Set of such mappings.

Let $S \in \mathbb N$. Let $\mathfrak S_S$ be the set of permutations of size $S$. Consider map $f : \mathfrak S_S \times \{1,2,\ldots,S\} \to \mathfrak S_S$, such as $f(\alpha, \cdot) : ...
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41 views

For which integers $r$ is $\sigma ^r$ also a $k$-cycle? [duplicate]

Let $\sigma$ be a $k$-cycle in $S_n$. For which integers $r$, is $\sigma ^r$ also a $k$-cycle? I think I managed to prove that this is true iff $(k,r)=1$, but my proof was too long and not elegant ...
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117 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
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90 views

Hashing: Quadratic Probing

I have the following to prove, unfortunately I am not able to do so. Let h, h' be hash functions: $h(k,i) = (h'(k) + c_{1}i + c_{2}i^2)$ mod $m$. Show the following: if m is prime and $c_{2} \neq 0$ ...
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100 views

What is the distribution of cycle lengths in derangements? In particular, expected longest cycle.

There is a lot of information about expected cycle lengths in random permutations, but I'm having trouble adapting the arguments and calculations to the specific case of derangements - permutations in ...
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70 views

To prove an identity in permutation and combination.

I am trying to prove the following identity: ${n \choose 0}$ + ${n \choose 1}$ + $\ldots$ + $\frac{1}{2}{n \choose n/2}$ = $2^{n-1}$ where $n$ is even I know that I have to use few relations like ...
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1answer
62 views

Disjoint Cycles in a Cyclic Subgroup of $S_n$

If a permutation $\sigma$ $\in$ $S_n$, the permutation group of n elements, and $\sigma$ can be expressed as a product of disjoint cycles, is it necessary that the disjoint cycles be elements in ...
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243 views

In how many ways can a natural number be written as a sum of $2$ natural numbers?

For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.
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60 views

Random permutation and isolated points on the line

Let $[n]=\{1,\dots,n\}$ be the (ordered) set of the $n$ first integers, and $\mathcal{S}_n$ denote the set of permutations of $[n]$. Let $1\leq k \leq \frac{n}{4}$ be an integer. If I draw uniformly ...
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29 views

Success rate of a player trying to guess a bitstring with given constriants

For work at my university I try to solve a problem. I have a bit string with given length $len$ and count of active bit $active$ An example could be: 1001 0110 ...
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136 views

Prove a cycle of length l is odd if l is even? [closed]

This is my first course on Group Theory. How do I go about proving this?
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87 views

Number of ways arranging entries of a tuple - combinations or permutations

Let $x=(x_1,x_2,\ldots,x_n)$ be an $n$-tuple where $n$ is even In how many ways we can arrange such that exactly half of the entries are even ? My attempt is : As we are talking about possible ...
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3answers
67 views

find all odd permutations $\sigma \in S_4$ such that $\sigma (123) \sigma^{-1} = (234) $

need help with this question... find all odd permutations $\sigma \in S_4$ such that $\sigma (123) \sigma^{-1} = (234) $ really have no idea how to approach this. thanks.
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1answer
37 views

Finding the number of combinations

A teacher distributes 7 books to 7 children (each student a books), on the next day she collects the books back and redistributes in such a way that each students get a new book. In how many ways can ...
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1answer
442 views

Find the probability of at least two vowels together when letters in word “AEINCB” are rearranged for all random permutations.

Find the probability of at least two vowels together when letters in word "AEINCB" are rearranged for all random permutations. What will be new probabilities when word is changed to either "AEINCBB" ...
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53 views

Can anyone explain these conclusions? Permutations, Symetric group…

The conclusions start off like this:I will highlight what is unclear in yellow. $sgn G$-sign of G permutation, $Ker$-kernel of a function Lets define the function: $\ \Phi$ like: $(\forall G \in ...
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75 views

Finding the number of ways of picking three cards

Problem: An urn has 10 red cards numbered 1 through 10 and 8 blue cards numbered 1 through 8. Three cards are randomly drawn, one at a time, without replacement. Find the number of ways to ...
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100 views

Finding how many bits of length n there are

So we are starting on the section of combinatorics in my discrete math class and our instructor gave us a simple problem to see if we understood what we learned that day. The problem consists of three ...
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90 views

question based on probability/permutation/combination

In a box containing $15$ apples, $6$ apples are rotten. Each day one apple is taken out from the box. What is the probability that after four days there are exactly $8$ apples in the box that are not ...
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1answer
131 views

Sum of Permutations of n objects taken r at a time ( r=1 to n ) where objects may be groups of same entities

Given n objects where n1 objects are the same ,along with another group of n2 objects of same element etc.. such that Σni = n (i=1 to k). Assuming there are k groups of similar objects eg: in ...
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111 views

Sum of all the numbers with the given numbers repeated

How to find the sum of all the numbers that can be formed using the digits 4,5,5,6,6,6 (This includes 4,5,6,45,46,54,55,....,666554). I knew that the answer is 39345806. I just need to know the method ...
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3answers
280 views

Permutations without repetitions (exclude repeated permutations) [duplicate]

The formula to calculate all permutations without repetitions of the set {1,2,3} is $\dfrac{n!}{(n-r)!}$ But how to calculate it if the set (or rather array in programming terms) includes repeated ...
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55 views

“quasi-increasing” permutation of a number

Call a permutation $a_1,a_2,\ldots,a_n$ quasi-increasing if $a_k\le a_{k+1}+2$ for each $1\le k\le n-1$. For example, $54321$ and $14253$ are quasi-increasing permutations of the integers ...
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24 views

Number of permutations with balanced middle element

Let $v$ be a permutation of $\{1,2,\cdots,2n+1\}$ where $n$ is odd, such that the middle element $v_{n+1}$ satisfies the following: the number of elements to the right of $v_{n+1}$ that are less than ...
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63 views

How many arrangements of 4 letters, with 3 of them being distinct, are there?

I read an example of the "Counting Principle" where we want to find the number of possible ways to rearrange 4 distinct letters chosen from the alphabet. The answer for this one makes sense. This is ...
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31 views

Arranging $2n$ objects in specific ways

There are $n$ objects $a_1, a_2, ... , a_n$ and another $n$ objects $b_1, b_2, ... , b_n$. We have to choose all the $2n$ objects such that $a_i$ is chosen before $a_{i+1}$ and $b_i$ is chosen before ...
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1answer
42 views

How to finding permutations where some elements repeat?

Sorry if my question is not mathematically correct. Please help me fix it if there is a better way to phrase it. So first of all, I know that if you have a list of numbers {1, 2, 3} then the number ...
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3answers
61 views

Permutation of Numbers

How many $5$ digit numbers can be formed from the integers $\{1,2,...,9\}$ if no digit can appear more than twice.(for example 41434 is not allowed) My approach is : Since, $max $ 2$ digits can ...
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2answers
44 views

Permutation of Indistinguishable Objects

How many number of two digit numbers can be formed using $\{4,5,6,6\}$ without repetition? I know that $\{45,46,54,56,65,64,66\}$ are the possible answers, but I am wondering if there is any formula ...
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271 views

Sum of permutations of a number

If I have a number (without any 0's) such as 1112334, how would I sum the permutations of its digits (excluding duplicates)? I am assuming there is a closed form involving factorials or combinatorics. ...
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52 views

Find the order of $\tau^{100}$

Let $\tau= \left( \begin{array}{ccc} ...
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42 views

Another kind of derangement?

I reading about derangements, and the following question came to my mind. Suppose in an office, there work 5 teams, each consisting of 1 head and 3 staff (so there is a total of 15 staff). If the ...
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1answer
35 views

Permutation of kronecker products

I would like to be able to compute a re-ordered kronecker product from the result of another kronecker product. For example, consider $$F=A\otimes B\otimes C\otimes D\otimes E$$ from the result F and ...
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The amount of unique structures that can be built from four different bases, without mirrored repetitions.

I'm trying to figure out the following problem. I have 4 different bases (A,C,G and T), and I'm trying to figure out how many possible unique structures I could build with them with different sequence ...
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68 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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31 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
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187 views

How many permutations of the letters DANIEL do not begin with D or do not end with L?

How many permutations of the letters DANIEL do not begin with D or do not end with L? The correct answer is 696. This answer does not make sense as there are 120 (5!) ways the letters can be ...
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93 views

In how many ways can the committee be selected if the girls must include either Roberta or Priya but not both? [duplicate]

A committee of three boys and three girls is to be selected from a class of $14$ boys and $17$ girls. In how many ways can the committee be selected if the girls must include either Roberta or Priya ...
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256 views

Permutations and Combinations Tricky Question

In a photo there are three families (six Greens, four Browns, and seven Grays) arranged in a row. The Browns have had an argument so no Brown will stand next to another Brown. How many different ...
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Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
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223 views

How many ways are there of splitting twelve people into two groups of the same size?

Twelve people need to be split up into teams for a quiz. How many ways are there of splitting them into two groups of the same size? I did $12 C 6$, which gives $924$, however the answer is ...
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2answers
339 views

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet?

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet? I tried $5 C 1 + 5 C 2 + 5 C 3 + 5 C 4 = 30$, however, the answer is $20$. Any ...
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114 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
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Combination or Permutations - identification

Car $A$ can take $5$ passengers, car $B$ can take $6$ passengers, car C can take $2$ passengers. Find the number of ways that $11$ passengers and a couple to travel in the $3$ cars. $${13 \choose{5}} ...
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28 views

Inverse Permutations from $S_7$

Would someone mind giving an explanation of how to find the inverse permutation of: $(1 2 3 5 7)^{-1}$ in $S_7$? I am not quite understanding how to do this.
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133 views

Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
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70 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
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1answer
66 views

Best algorithm for finding permutations with constraint of average total value.

Let's assume I have a random number generator from 0-100 included (only integers) and I generate 5 numbers with it. I want to know the probability of hitting 80, 80, 80, 80, 80 with the constraint ...
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83 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
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59 views

Permutations for a set of rules

The question is from - http://www.iarcs.org.in/inoi/2015/zio2015/zio2015-question-paper.pdf - Q.2 I tried solving it but I have no clue how to go about doing it. The question says that a railway ...