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

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Automorphism group of the Alternating Group - a proof

I was trying to read the following lemma which admit as an easy corollary the structure of the automorphism group of the alternating group on $n\geq 7$ elements. Anyway there are two points that ...
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permutations with the English alphabet

How many four-letter words, using the English alphabet, are possible if letters if only vowels may be repeated? How many four-letter words are there if at most one repetition of any letter is allowed? ...
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A rental car agency has 12 identical cars available and 7 identical vans…

My question is: A rental car agency has 12 identical cars available and 7 identical vans a) If the group needs to rent four cars and two vans, in how many different ways can they select their ...
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to find total number of subsets

I was working out some problem where I needed permutation and combination. I took the cartesian product of $n$ sets where number of elements in each set is even and $n$ is odd. Further the elements of ...
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Combinatorics question with stars and planets [closed]

Assume that a small universe has 10 distinct stars and 100 distinct planets so that 20 of them are habitable and 80 of them are nonhabitable by humans. How many ways are there to form a galaxy with ...
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$D_6$ and cycle notation problem

I have a hexagon with edges $A,B,C,D,E,F$ and I want to work with its symmetry group $D_6$ in cycle notation. My calculations don't yield consistent results. For example, I correctly get $r^4 \cdot ...
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Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$…

Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$,while $4$ appears in a different cycle of $\pi$ from $1,2,3$? My attempt:For $n=4$ I ...
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41 views

Calculating possible combinations for 1-3 digit code

I have been researching a lot on permutations and calculating total numbers of combinations of certain array lengths allowing certain characters. However, all the equations used to do this only ...
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One-To-One functions

Let A be the set with n elements and B be the set with m elements. How many one-to-one functions are there from A to P(B) (power set of B). There are n! total functions from A to B. and (2m)n from A ...
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25 views

Symmetries of a $9$ puzzle (Rubik's Slide)

Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also ...
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Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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How many 2m-permutations, consisting only of cycles of even length?

How many 2m-permutations, consisting only of cycles of even length? I have found this formula: $$Q_2(n) =((2n − 1)!!)^2$$ but how it can be proven?
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fixed length of permutaions cycles

How much permutations has only 10 cycles, but three of them has length 3 and seven of them has length 7?
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14 views

Permutation test and p-value

I construct a permutation test in order to see If two samples come from the same distribution or not. I have two vectors $x, y$ that hold values of sampled values from two populations and the test ...
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47 views

Problem Solving Involving Permutation

Find the number of 6-digits number with no 3 consecutive number with same digits. Note that 0 might be the first number. I have tried to find the number with no pairs, 1 pairs, 2 pairs and 3 pairs. ...
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example for permutizer group

permutizer of a subgroup H of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with H. You can help us give an example?
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Multi-ruled combinatorics problem (need this for my lab)

I need to know this for practical purposes and not homework, learning etc.. Say I have 3 electrodes A,B and C. Say I also have 3 electrolytes A,B and C. If electrode A has to be in electrolyte A, ...
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let $D_n$ be the number of permutations of $\{1,2,3,…n\}$ which leave no element fixed.

Let $n\geq2$ and let $D_n$ be the number of permutations of $\{1,2,3,\dots,n\}$ which leave no element fixed. How to write an expression for $D_n$ in terms of $D_k$? I don't know how to start. Please ...
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Unique permutations from set with repetitions

I am new to combinatorics and might ask a trivial question: There are $69$ different items, each present $4$ times. From this total of $276$ items, $20$ should be picked at random. I need the formula ...
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Number of lists at some Kendall-Tau distance

Given a ranked list (permutation) $R$ of $n$ elements, how many permutations of the same elements are there at Kendall-Tau distance $d$ from $R$ $(0 \le d \le \frac{n(n-1)}{2})$? Example: If $R = ...
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Possible 4 character passwords involving a letter and a digit.

A password consists of 4 characters, each of which is either a digit or a letter of the alphabet. Each password must contain at least ONE digit and AT LEAST ONE letter. How many different such ...
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How many permutations do we need before we're in $SU\left( n\right)$?

Let $\mathcal{L}\subseteq \mathfrak{su}\left( n\right)$ be a Lie algebra for $n \geq 2$ with Lie group $G = e^{\mathcal L}$, and let $X \in G$ be represented by an $n\times n$ matrix (I prefer fixing ...
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34 views

How many different teams can be created between two groups?

If a company has 8 painters and 12 electricians. How many different teams can be created with 1 painter and 1 electrician? I know that the number of ways a team can be made is: $ {8 \choose 1} * ...
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Problem regarding proving a permutation group

The question states: Show that the set of permutations of three objects form a group. Give the multiplication table for this group. If we take three distinct objects, the set of the ...
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compute the number of permutations

Compute the number of permutations of $\{1,2,3,4,5,6,7,8,9\}$ in which either $2,3,4$ are consecutive or $4,5$ are consecutive or $8,9,2$ are consecutive. I know we will use some exclusion-inclusion ...
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Question about permutation.

Suppose a and b are permutations of the same cycle type. Why aligning them on top of one another and interpret it as a two line representation of permutation gives me a permutation that will conjugate ...
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In how many ways 3 persons can solve N problems.

There are $3$ friends $(A,B,C)$ preparing for math exam. There are $N$ problems to solve in $N$ minutes. It is given that: Each problem will take $1$ minute to solve. So all $N$ problems will be ...
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1answer
45 views

What is the probability of not rolling any given number on 10 rolls of a die?

In other words, ALL combinations which don't contain at least one of the number from 1-6 would count. So for example... 5, 2, 3, 3, 4, 1, 5, 5, 3, 1 would be counted because there is no 6 Also 5, ...
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Better Explanation for an already posted question [duplicate]

Can anyone explain why in this question the answer is 5! * 2! * 10P3? I understand the 5! and 2! but for 10P3 the first thing I thought of was 3! Thanks.
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Permutation Question Help

Hexadecimal numbers are made using the sixteen digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. They are denoted by the subscript 16. For example, 9A2D$_{16}$ and BC54$_{16}$ are ...
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The order of a cyclic subgroup, generated by a permutation

I was wondering, how can I prove that all cyclic subgroups generated by a permutation, has the same order as the permutation? For example, cyclic subgroup $\langle(---)\rangle$ will have order 3. So ...
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Number of 8 character passwords including numbers and letters without repetition

A password must be created with 8 characters. It can use number or letters, but they cannot be repeated (and letters are not case sensitive so we have only 36 characters). How many passwords are ...
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34 views

Calculating the probability of receiving all possible rewards after 15 events

I encountered this question in my Data Management and Statistics textbook. I tried to calculate the probability using binomial theorem and combinations/permutations, but I could only get close to the ...
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what is the probability that the real estate agent can get into specific home ???

A real estate agent has 8 master keys to open several new home. Only 1 master key will open any given house. If 40% of the homes are usually left unlocked what is the probability that the real estate ...
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How many different ways can a student check off one answer to each question?

If a multiple-choice test consists of 6 questions each with 4 possible answers of which only 1 is correct, In how many different ways can a student check off one answer to each question ?
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A question concerning 4-cycles in $S_4$

Is it true that for all $g\in S_4$ and $f \in S_4$ a 4-cycle, then $g^{-1}fg=h$ implies $h$ is also a 4 cycle. I did a few examples, and it seems to be true, but I don't know how to prove it. Also, I ...
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Counting number of ways in poker game

What is the total number of ways in which the poker hand is full of house that is you have to pick 5 cards out of 52 cards such that it contains exactly 3 cards with the same value. Example a card ...
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Cycle structure in a symmetric group

I have a bit of a problem. I'm currently reading about permutations, and I have a little exercise that asked me to find all cycle structures in $S_6$. I came up with the following $ ( -)\\ (- -)\\ (- ...
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$G$ is a primitive group

Let permutation group $G$ contains a minimal normal subgroup $\neq 1$ which is transitive and Abelian. Show that $G$ is primitive. My attempts: Because of Proposition 4.4. of Wielandt's book ...
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Permutation of Groups - looking for the right term

I'm looking for more detailed information about the following problem, but i'm missing a right keyword, or term for this: Let's assume i have 10 people and they are assigned to groups: ...
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Represent a bijection using a permutation

Let $X = \{1, 2, 3, 4, 5, 6, 7\}.$ For every $n \in X$, write $n^2 - 3n^5 = 7q_n + r_n, 1 \leq r_n \leq 7.$ Define a function $f: X \to X$ by $f(n) = r_n.$ (a) Find an element $\alpha \in S_7$ that ...
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Arranging books on the shelf.

There are five distinct computer science books, three distinct mathematics books, and two distinct art books. In how many ways can these books be arranged on a shelf if no two of the three mathematics ...
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1answer
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Select K numbers from N numbers fairly

I want to fairly select K numbers out of an array of N number. I know that this problem can be solved using Reservoir Sampling but I want to know if this approach is correct too? ...
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41 views

Permutation (inclusion-exclusion)

2 corrected exams are being returned to each of n students. How many ways can the teacher give those 2 exams back to each student such that everyone receives at least 1 exam that is not his. I know ...
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How many sequences of length N squared can be formed with N different values where each value is used exactly N times?

For instance, for N=2, the answer is 6 (e.g. aabb, abab abba baab baba bbaa). For N=3, the answer is 1680. I'm looking for the proper formula. Thanks
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Is there an effective way to convert a product of 2-cycles into a product of n cycles?

I came across this problem that asks me to convert (12)(34) into a product of 5 cycles. After testing for many different combinations i get (12345)(14352)(12345). The way I do it is this: ...
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Strategy for number of non-negative integers solutions such that $x_1+x_2+\frac{\enspace\enspace\enspace}{}+x_5 = 50$

I'm trying to figure out the number of solutions to the following problems, although I'm not entirely sure what strategy I should use to solve these. Combinations of non-negative integers ...
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Permutations and school timetable

If there are 6 periods in each working day of a school. In how many different ways can one arrange 5 subjects such that each subject is allowed at least one period? I tried this way- One of the six ...
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Permutations in products of disjointed cycles

How do I calculate the following permutation in the symmetric group $S_6$ giving the answers as products of disjoint cycles: $$(2,3,5,6)(1,6,2,4)$$ I have tried following this question but I don't ...