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

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The number of ways of dividing a number by three separate integers.

How many ways can I arrive at the number $45$ by exactly using $5$, $10$ and $20$. I can use each number as many times as necessary. (e.g $9×5$, $20+(5×5)$) this leads to the question, if the number ...
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1answer
50 views

How to interpret combination and permutation problems?

This is more of a methods question than asking for a specific answer: In revisiting statistics and attempting various problems, I am curious if anyone has any insights on how to "see" the route to ...
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1answer
21 views

Multiple Group Representations using Cayley's Thm

I know that an abstract group can be made isomorphic to a subgroup of a symmetric group, by using a Cayley table for that abstract group. However, what is a technique for getting another permutation ...
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2answers
39 views

maths permutation help

An experiment consists of randomly rearranging the 9 letters of the word TARANTULA, where all possible orders of the 9 letters are equally likely. Find the probability of each of the following events: ...
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1answer
31 views

Permutations and group acts

How many ordered pairs of permutations $(\pi , \sigma )$ in $S_n$ such that $\pi \circ \sigma =\sigma \circ \pi $. I think i need consider group acts on itself by conjugation $\pi (\sigma )=\pi \circ ...
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1answer
46 views

Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states: Let $G$ act ...
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1answer
125 views

problem on permutations

In $S_{10}$, can someone explain why there is no permutation $a$ such that $a(1,2,3)a^{-1} = (1,3)(5,7,8)$?
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4answers
169 views

Find the number of ways to form 15 teams out of 15 men and 15 women.

In how many ways can 15 teams be formed, each consisting of a man and a woman, from 15 men and 15 women. This looks like the same problem as finding the number of bijective functions from a set $A$ ...
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1answer
30 views

What is the process behind finding a Cayley permutation representation.

For example, let's find the Cayley permutation representation of $\mathcal D_3$ in $S_6$. $\mathcal D_3 = \left<r,s \mid r^3=s^2=1, rs=sr^{-1}\right>$. Write, \begin{pmatrix} 1 & 2 & ...
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2answers
69 views

How do I find the permutation with the highest order in a symmetric group?

My professor gives this text, but I don't understand what it's saying, could someone explain it to me? Let $M(n)$ denote the largest order of an element in $S_n$. By Theorem 1 $M(n)$ is the largest ...
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1answer
16 views

A low-discrepancy or quasirandom series which would guarantee all value sequences

I am trying to find a type of quasi-random sequence which would guarantee that it could produce all possible sequences of values within the possible value range, while still producing random-seeming ...
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1answer
43 views

Permutations regarding $3$ women and $4$ men

$3$ women and $4$ men are standing in a line. If no two women may be adjacent to each other, how many distinct line-ups are there? I'm not sure how to do this. I know $4! = 24$ and $3! = 6$. Where ...
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2answers
98 views

Formula to calculate password cracking time in years, taking into account Moore's law and known adversary guessing power [closed]

We know that the biggest human rights violators in human history are capable of one trillion password guesses per second as of approximately January 2013. Assume that the 1 trillion guesses per ...
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1answer
111 views

Maximum number of points of intersection

The greatest number of points of intersection of 8 straight lines and 4 circles are? My attempt:Assuming every line cuts all the four circles at two points each, the points of intersection of lines ...
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1answer
31 views

Ways of forming a committee

Four couples (husband and wife) decide to form a committee of four members.Find the number of different committees that can be formed in which no couple finds a place is? My attempt:In one case where ...
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1answer
50 views

To find number of questions when number of wrong answers is given

In a certain test there are n questions, in this test $2^{n−i}$ students gave wrong answer to at least i questions where i=1,2,3,…,n. If the total number of wrong answers given is 2047,what is the ...
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0answers
33 views

Recursive formula for the number of $n$-permutations with $k$ cycles

Let $n$ and $k$ be a positive integers satisfying $n\geq k$, then $$c(n,k)=(n-1)c(n-1,k)+c(n-1,k-1)$$ where $c(n,k)$ denotes the number of $n$-permutations with $k$ cycles. The proof of this ...
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2answers
38 views

No of ways of making a selection.

There are $n$ different books and $p$ copies of each. Find the number of ways in which a selection can be made. My attempt: When he says, make a selection, I assumed that one book is to be ...
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1answer
49 views

To find number of questions in a test when number of wrong answers is given

In a certain test there are $n$ questions, in this test $2^{n-i}$ students gave wrong answer to at least $i$ questions where $i=1,2,3,\ldots,n$. If the total number of wrong answers given is ...
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1answer
45 views

Smallest value of n to form 900 n-digit numbers using given digits [closed]

An $n$-digit number is a positive number with exactly n digits. $900$ $n$-digit numbers are to be formed using only $2$, $5$ and $7$. What is the minimum value of $n$ for which this is possible? ...
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1answer
30 views

Ways of dividing people into groups.

The number of ways of dividing $12$ people into $3$ groups of $4$ each is? My attempt:First we choose $4$ members in $(12C4)$ ways.And then out of remaining $8$ we choose them in $(8C4)$ ways.Finally ...
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1answer
58 views

Permutations of Characters

How many strings of five characters use the letters a, b, c, d and do not have the letters bad appearing consecutively and in this order? (Letters can occur any number of times, including not at all.) ...
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19 views

Simple lemma about permutations

While doing some recalling about permutations I've crossed with the following simple lemma: Let $g:[n]\to [n]$ be a permutation. Let $x\in [n]$, and there exist $1\leq i\leq n$ so that $g^i(x)=x$. ...
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62 views

problems on permutation

There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of ...
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60 views

Elusive closed form for card permutation problem

Does a closed form formula f(n) exist for the two rightmost columns? The two question marks are meant to be 0. The diagram is a summary of the numerical results from original question: Permutations ...
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1answer
22 views

Possible Ecommerce Product, Size, and Option combinations

I have an ecommerce site where you can specify the size of a product and any number of options depending on the different products. Each option has a category. For example, for each product you can ...
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1answer
62 views

Does anybody spot anything familiar in this integer sequence?

$0,3,9,21,40,67,106,154,220,298,395,510,644,\dots$ These are the maxima of the distances between permutations of length $n$ up to $n=13$ according to a modified version of Spearman's footrule number ...
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1answer
27 views

How are the equivalence classes made of orbits defined

I am reading through Freileigh, and I'm curious to know if I am understanding the definition correctly. Each Permutation of a set $A$ determines a natural partition of $A$ into cells with the ...
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0answers
61 views

Counting all permutations(repeating) with no adjacent elements equal and m majority elements.

Counting all permutations(repeating) of 1 to n which have size n. 1. with no adjacent elements equal, and 2. m majority elements(A element is majority element when it appears max number of times in ...
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1answer
34 views

Number of permutations with subset distance constraint

The problem is to calculate the number of all unique permutations of a string with repetitions. There is also a constraint for one subset elements to be spaced from each other. Typical input data is ...
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1answer
16 views

Number of permutations on a lattice

Let's say I have a lattice like that: It's a lattice 10x10 (or N*N). So 100 little squares compose the lattice. Now I have to put 6 green squares (or n green squares) on the lattice. How do I ...
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4answers
109 views

How to visualize permutations?

I'm getting a warning that this is a subjective question, and it very well probably is. But nevertheless, it is still a valid question that helps in the studying of mathematics from my point of view. ...
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2answers
28 views

Discrete Math Combinatorics, permutation, one-to-one proof

I am having trouble getting started with the following proof: (This is homework, so I'd appreciate a nudge in the right direction.) Let m, r $\in$ N with 0 $\leq$ r $\leq$ m. Prove that the number of ...
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1answer
27 views

Method for finding permutation of n elements if you have all permutations of (n-1) elements

In the "The Art of Computer Programming Volume 1 third edition " chapter 1.2.5. Permutations and factorials professor Knuth introduces method for constructing all permutation of $n$ objects from ...
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Proving that matrices $B$ produced by transposing two rows of $I$ satisfy $B=B^{-1}$?

For example, I have a $3 \times 3$ identity matrix. If I exchange rows $2$ and $3$, then I get $$ B = \pmatrix{1&0&0\\0&0&1\\0&1&0}. $$ In this case it can be checked that ...
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Permutations of k types for a number smaller then total of k's (without replacement)

How can you find the number of permutations of a set of items grouped in different categories when you must choose less than the total of the set. For example, I have a set of $12$ songs that are in ...
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1answer
42 views

How many inversions in a permutation?

For the permutation $\begin{pmatrix}5&2&4&3&6&1\\4&1&3&2&6&5\end{pmatrix}$, determine if it is even based on its inversions. So here's my trouble; I've worked ...
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2answers
32 views

Probability question on finding number of ways to label all boxes incorrectly with respect to the color of the ball they contain

Question: There are balls of 5 different colors - yellow, blue, red, green and white. A worker has to separate these balls as per their colors into different boxes and label them with corresponding ...
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93 views

permutations of length N of N natural numbers from 1,2,3…N such that no two consecutive numbers are equal.

For given n, I need to find sum of count of most occurring number for each permutation of length n.(such that no two consecutive numbers are equal). for e.g. for n=3 1 2 3 will give 3 as the count ...
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1answer
38 views

Transposition as generating $S_n$.

As far as I know the group of symmetries of the euclidean n-cube is generated by reversions ${r_i}$ and transpositions ${s_i}$ for i=1,...,n-1. Transpositions generate the subgroup of permutations of ...
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56 views

the number of inversions in the permutation “reverse”

Known, that number of inversions is $k$ in permutation: $$\begin{pmatrix} 1& ...& n& \\ a_1& ...& a_n& \end{pmatrix}$$ Find number of inversions in permutation (let's ...
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50 views

Number of combinations of k types from set n with limited number of each type

How can I find the number of combinations of types from a set of multiple types when the number of each type is limited? For example, I have a set of 3 chicken dishes, 4 beef dishes, and 5 lamb ...
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Combinatorics : number of non-decreasing series of r distinct numbers where the size of series ranges from 1 to N

I understand that number of non decreasing sequences of size M with N distinct numbers is (N+M−1)C(M). However, I'm interested in finding out the number of such series of r distinct integers where the ...
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1answer
33 views

Determine if a lottery system is profitable

I have a problem where I'm supposed to determine if a lottery system is profitable. I solved the problem and found it to be profitable, but I am not 100% sure about all of my calculations. Below is ...
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2answers
80 views

In how many ways can you distribute $3$ chocolates among $2$ kids if you have to give all $3$ of the chocolates to the kids? [closed]

In how many ways can you distribute $3$ chocolates among $2$ kids? One kid can get none. But we need to give away all $3$ of the chocolates to the $2$ kids. Why is it wrong to use $3C2$(which gives ...
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2answers
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Normalizer of the cyclic group in $S_n$

Let $G = S_n$ and $H = \langle (1,2,\ldots,n) \rangle.$ It is not too hard to see that $$C_G(H) = H.$$ What I am now wondering is, which group is $N_G(H)?$ Is there any way to determine that? I ...
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2answers
79 views

Expected value of random permutation

Let n ≥ 1 be an integer and consider a uniformly random permutation $a_1$, $a_2$, . . . ,$a_n$ of the set {1, 2, . . . , n}. Define the random variable X to be the number of indices i for which 1 ≤ i ...
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1answer
36 views

Cyclic type of the product of two permutations

Maybe this question is way too simple but I'm stuck. Suppose $x,y \in S_n$ and the product $xy$ has the cyclic type $(t_1,t_2,\ldots,t_n)$; here $t_i$ is the number of cycles of length $i$ when we ...
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1answer
15 views

permutations repetitions/no repetitions

License plates consist of sequence of 3 letters followed by 3 digits. How can they be arranged if (i) no repetition of letters is permitted, how many possible license plates are there? should it be ...
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Permutations : If repetitions are allowed

For example if a question is to find the number of different ways of arranging $4$ letters of $26$-letter alphabet with repetition, I know that we have to do $26^4$. However, I am confused as to why ...