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

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Conjugate cycles

Prove the following in $S_n$: Let $\alpha = (a_1,a_2,\ldots,a_s)$ be a cycle and let $\pi$ be a permutation in $S_n$. Then $\pi\alpha\pi^{-1}$ is the cycle $(\pi(a_1),\ldots,\pi(a_s)).$ I'm not ...
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1answer
31 views

Cycles “converging” to an infinite cycle?

I recently had as an assignment, to find cycles $\sigma,\tau\in S_{\mathbb{N}}$ (i.e. permutations over the naturals) such that $ord(\sigma)=ord(\tau)=2$ and $\tau\circ\sigma$ has order infinity. This ...
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1answer
109 views

Mary can answer 20/25 problems correct… simple probability

Question: A teacher gave his class $25$ problems and told his students that he would select $10$ of them to put on their midterm. Mary can figure out how to answer $20$ of the problems, what is the ...
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1answer
201 views

What's the difference between a permutation and a combination with repetition?

My understanding is that a permutation is used to find the number of rearrangements of different elements, taking into account possible orders. A combination is used to find the number of ...
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112 views

Multiple Hypergeometric Distributions

I need to figure out a problem which involves multiple hypergeometric distributions. Referring to the Urn problem, the problem can be described like the following: We have $n$ urns $u_1,…,u_n$. Urn ...
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1answer
118 views

Using combinatorial reasoning to show $n!=\binom{n}{0}D_n+\binom{n}{1}D_{n-1}+\dots+\binom{n}{n}D_0$

How can one use combinatorial reasoning to show that $$n!=\dbinom{n}{0}D_n+\dbinom{n}{1}D_{n-1}+\dbinom{n}{2}D_{n-2}+....+\dbinom{n}{n-1}D_1+\dbinom{n}{n}D_0$$ Now $D$ stands for deranged which is a ...
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1answer
135 views

Largest Number Drawn - Why are These Approaches Not Equivalent?

Here's the question: Four numbers are drawn at random from a box of ten numbers 0, 1, ..., 9. Find the probability that the largest number drawn is a six if the draws are made with replacement. The ...
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4answers
92 views

determining the amount of total questions needed in a game given the probabilty

I'm creating a game and can't seem to quite figure this out - driving me crazy. There are 8 questions in my game You can play the game an unlimited amount of times the test bank doesn't change. so ...
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2answers
1k views

Permutation and combination problem - word arrangement

This is a question of permutation and combination. Q. How many words can be formed from the word "LUCKNOW" when i) No restriction is there ii) L is the first letter of the word iii) All the ...
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136 views

How find this $aA_{m+1}=\overline{\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{m}}$

Question let $m$ is positive numbers,and such $m\ge 5$,and $$A_{m+1}=\overline{1234\cdots m}=1\times (m+1)^{m-1}+2\times (m+1)^{m-2}+\cdots+(m-1)\times (m+1)+m$$(or see ...
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1answer
744 views

In how many ways can you arrange the alphabet so that A and B are always next to one another (In either order)

Ok, so I have no idea where to begin on this question. Do I treat A and B as one letter? 25 choose n?
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1answer
114 views

How many ways are there to arrange 1's and 0's with no two 1's in a row? [duplicate]

Given n spaces, how many ways are there to fill up the spaces with 1's and 0's such that no two 1's are together. For example, let's say n = 3 (_ _ _). There are 5 ways to fill up the spaces such ...
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1answer
158 views

Calculate total number of combination of 4 characters having pattern as Letter-Number-Letter-Number

I need to know "How" to calculate total number of combinations that are possible to generate 4 character string having a pattern of Letter-Number-Letter-Number. The complexity are: strings should be ...
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1answer
33 views

What are the number of circular arrangements possible?

Suppose we have $4$ identical red beads and $3$ identical blue beads. In how many ways can we form a necklace out of these? I am a little confused here. Suppose we fix a red bead and treat the ...
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1answer
63 views

Finding $\lim\limits_{n\rightarrow \infty}\sum\limits_{r=1}^{n}\frac{1}{T_r}$ given $\sum\limits_{r=1}^{n}T_r=\frac{n(n+1)(n+2)(n+3)}{8}$

If $\displaystyle\sum_{r=1}^{n}T_r=\frac{n(n+1)(n+2)(n+3)}{8}$, then how can we find $\displaystyle\lim_{n\rightarrow \infty}\sum_{r=1}^{n}\frac{1}{T_r}$?
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2answers
54 views

Permutation help

Consider the elements of $S_7$. For each $\sigma \in S_7$ there is a smallest positive integer |$\sigma$| such that $\sigma^{|\sigma|}=e$. Find the value of $N$= max{ $|\sigma|$ | $\sigma \in S_n$}. ...
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1answer
44 views

Finding a permutation from a power of itself

Find a permutation $\sigma \in S_9$ such that $\sigma^2=(13579)(268).$ So I know that $\sigma^{10}=\sigma.$ But I don't know $\sigma^5$..... Is $\sigma^{10}=\sigma^4\sigma^6$? I doubt this is the ...
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1answer
643 views

How many different ice cream cones with 31 different flavors and 2 kinds of cones?

I have been trying this for a while now. Using the formula for permutations, I am getting P(31, 2) = 10,230, but this seems way too high... An ice cream shop has 31 different flavors of ice cream and ...
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2answers
787 views

If a club has 24 members, In how many ways can 4 officers be chosen from the members of the club?

I understand the concept of combinations and permutations. However, I am not getting how to apply the formulas. I believe understanding exactly how to do this would help.A club has 24 members. a. In ...
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1answer
3k views

Permutations/Combinations: How many different passwords are possible?

Hello everyone. I have a couple questions this time, but I think if I understand how to do this one, I'll understand the others. A particular online banking system uses the following rules for its ...
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1answer
194 views

Alias/alibi in permutation groups

This question came up in teaching a course on basic group theory to high school students. I gave the class the task of enumerating as many subgroups as they could find in $S_4$ and in the group $O$ ...
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1answer
39 views

Selecting unique questions without replacement

I'm trying to create a Q&A game and have a question. X = number of questions (the question bank) I randomly choose 40 questions from X. A question is chosen with replacement. I want to figure ...
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4answers
288 views

Probability of two digit number sequence in series of numbers

Given a random sequence (say $15$) of numbers I want to find the odds of finding '$90$' and '$09$' in the sequence. Looking at just two numbers in the sequence you have a $\dfrac{2}{10}$ chance of ...
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1answer
79 views

Signature of permutations is a homomorphism

Given the following definition of $signature$: $\epsilon(\sigma)=(-1)^{n-k}$, where $k$ is the number of cycles (with disjoint supports, counting the 1-cycles) of the permutation, prove that ...
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2answers
78 views

What is the centralizer of (1 2 3)(4 5 6) in $S_6$

So far, I've seen that the following permutations are in the centralizer: $(1 4), (2 5), (3 6)$, products of these transpositions(EDIT: not all of these are in the centralzier), $(1 2 3), (1 3 2), (4 ...
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203 views

The number of ways to get N as the sum of R elements with constraints

The number of ways to get N as the sum of R elements, except my solution must have no repetition (3+2 and 2+3 counts for only 1), and 0 cannot be used. For example: N=8, R=2, should return 4. The ...
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0answers
80 views

Is there an easy proof for the classification of $6$-transitive finite groups?

For the background, see the post: Classification of triply transitive finite groups Thanks to the classification of finite simple groups (CFSG), we know that if $G$ is a finite $6$-transitive ...
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1answer
66 views

How do I find “at most” x bit strings of length 20?

I tried searching online and I found several examples of doing such problems, but I'm still not sure if I'm doing them correctly and would greatly appreciate some help! How many bit strings of length ...
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2answers
31 views

How many solutions

If I want to have company A owns company B and there are 195 countires in the world and company A and Comapny B don't have to be in the same country mow many different permutations/solutions are there ...
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1answer
55 views

Am I correct or is the solution correct? I am pretty sure I am the one that is correct.

John and Mary are members of a group of eight boys and two girls. In how many ways can a committee of five be chosen from the the same group of ten if either John or Mary is in it but not both? My ...
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87 views

Odd Permutations

Prove that the product of two odd permutations is even. I'm having a difficult time doing this in the general case. I have that if s is even, then $$\alpha = ...
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0answers
203 views

Classification of triply transitive finite groups

A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $X\setminus\{x\}$. Is ...
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120 views

Confusion question in Permutation and combination

Q: How many ways can 4 prizes be given away to 3 students, if each boy is eligible for all the prizes? Ans: ...
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0answers
72 views

I want to generate(or count) all possible binary matrix that satisfy certain Condition

I want to generate(or count) all possible binary matrix that satisfy below Condition. let A be arbitrary binary matrix 4*4 ...
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1answer
95 views

What does permutation stand for as a power?

I am just reading some books about abstract algebra and I don't understand what a permutation stands for as a power. For example, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
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105 views

Permutation Problem with Given Numbers

Given the digits 0,2,5,6,9. A. How many 3-digit numbers can be formed if no two digits are to be the same? B. Of the numbers formed, how many are even? How many are odd? How many are greater than 600? ...
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30 views

Different arrangements - Permutation

Into how many different arrangements that look different can three identical trigonometry books, 4 identical calculus books, 5 identical algebra books be placed on a shelf?
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108 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...
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69 views

Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group ...
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1answer
322 views

How many positive integers n can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000?

According to my study guide the answer to the exercise, How many positive integers, (n), can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000, : The total of numbers n > ...
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1answer
68 views

How many different 5 characters words are there with only one letter a?

I just need to clarify my answer to this exercise. This is a permutations exercise. If we define a word to be a string of 5 letters of the English alphabet, regardless of meaning, then mnnnw is a ...
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42 views

Finding a permutation $ \alpha $ given $ \alpha^4 $ [duplicate]

I have the following question: Find a permutation $\alpha ∈ S_7 $ such that $\alpha^4 = (2 1 4 3 5 6 7)$. Is $\alpha$ unique? How should I go about this? I've tried a few different trial and error ...
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1answer
61 views

permutations and transpositions in even and odd cases

Say we had some $\sigma = (1, 2)(2, 3)...(n-1 ,n)$ could someone explain why this formula doesn't hold for odd n? For instance, $n = 2m+1$ $\sigma = (1,2)...(2m-2,2m-1)(2m, 2m+1)$, why does that not ...
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0answers
41 views

Product of permutations in $S_3$

Consider the elements $(1,2)(2,3)$ and $(2,3)(1,2)$ in $S_3$. We have that they equal $(1,3,2)$ and $(1,2,3)$ respectively. I am unable to make sense out of this. For the first product, let $\sigma = ...
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2answers
286 views

Counting four-digit numbers with repeating digits

Of all the four-digit positive integers containing only digits from the set $\{2,4,6,8\}$, what fraction of them have at least one of their digits repeating? Express your answer as a fraction. ...
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1answer
30 views

a question concerning subgroup of symmetric group

Suppose $H$ is a transitive subgroup of the symmetric group of $n$ symbols. Show that $n$ divides the order of $H$. I tried to show that some $n$-cycle is in $H$ but this idea did not work.
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4answers
841 views

How many of these numbers are divisible by 4?

There is this question that I have no idea where did I make the mistake. Each of the digits 1,1,2,3,3,4,6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit ...
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1answer
80 views

Combination Problem with mulitiple variables

I am new to this, but getting into math more and have a question regarding combinations and permutations with several variables involved. I work for a sales company and this question is based on ...
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118 views

Writing a Permutation as a product of Disjoint Cycles

Write the following as a product of disjoint cycles: $(1 3 2 5 6)(2 3)(4 6 5 1 2)$ I know from my solutions guide that the answer is: $(1 2 4)(3 5)(6)$ but I don't know how to do that. I started ...
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3answers
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A bowl contains 10 red balls and 10 blue balls, A women selects ball at random without looking?

How can we solve this question ? A bowl contains $10$ red balls and $10$ blue balls, and a women picks up balls from the bowl, at random, without looking. A) How many balls must she pickup in ...