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

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119 views

How to calculate a pair of cards contains at least one ace?

A pair of cards are simultaneously drawn from a deck of 52 cards three times in a row. The drawn cards are returned to the deck. What is the probability that two of three pairs contain an ace? For ...
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1answer
124 views

How many ways there are to arrange 40 people to play exactly one match each? [duplicate]

A tennis club has 40 members. They host a tournament playing single (one verses one) matches. Every member of the club plays one match with another member of he club, so twenty matches are ...
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4answers
622 views

Are there any Symmetric Groups that are cyclic?

Are there any Symmetric Groups that are cyclic? Because I have been doing some problems and I tend to notice that the problems I do that involve the symmetric group are not cyclic meaning they do not ...
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1answer
107 views

Nearest latin square

given a n x n matrix A with integer entries is there any way to find the nearest n x n latin square to it, say, e.g., in the Frobenius norm? I am looking for some type of convex optimization... ...
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2answers
379 views

Permutations to satisfy a challenging restriction

In a stack of n distinct cards in order {1,2,3,4,...,n} from top, define distance between 2 cards as the number of cards between them. 2 cards are neighbours if they're adjacent in original ...
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1answer
40 views

Find a subgroup of $S_{4}$ which is isomorphic to $\mathrm{Aut}(U_{8})$

The notation I am using is: $S_{4}$: the permutation group of order 4 $\mathrm{Aut}(U_{8})$: the set of all automorphisms on the set $U_{8}$ $U_{8}$: the group of numbers relatively prime to 8 I ...
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2answers
369 views

How many permutations of a multiset have a run of length k?

Background $\newcommand\ms[1]{\mathsf #1}\def\msP{\ms P}\def\msS{\ms S}\def\mfS{\mathfrak S}$Suppose I have $n$ marbles of $c$ colors, where $c≤n$. Let $n_i$ denote the number of marbles of color ...
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2answers
35 views

Arranging pictures possible combinations

I'm working on a problem which states there are 26 portraits of men and 4 of women. It wants to know how many ways can the photos be organized so no women are next to each other. I assume that the ...
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1answer
75 views

Dividing gems by random permutation

A group of people have found a treasure of gems: $G=90$ green and $B=990000$ blue. They decided to divide it among them. Since there are more people then gems, they decide to order themselves in a ...
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1answer
20 views

Group Theory - Permutations

If $B \in S_7$ and $|B^3| = 7$, prove that $|B|=7$. Solution: As $o(B^k) = o(B) / (o(B),k) $ Thus $|B| / (|B|,3) = 7$ Let $|B| = 7a$. Then $7a/(7a,3)$ should be $7a/a = 7$ or $(7a,3) = a$. As $3$ is ...
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1answer
66 views

Proof of 2^n deck of card, it will be reverse order performing n perfect in-shuffle.

I am now trying to prove performing n perfect in-shuffle with 2^n deck of card, and then it will be resulting reverse order. For example, Initial : [1, 2, 3, 4] 1st round : [3, 1, 4, 2] 2nd round : ...
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1answer
192 views

how many strings you can write with the letters abcd (permutation & combination or what?)

You have 4 letters abcd. How many 4-letter strings can you write with them? Assumptions: - the order is not important (aaab, abaa, baaa are same, counts 1) - you can use same letter more than once ...
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2answers
49 views

Permutations on a set with certain conditions.

Suppose we have a set $S=\{1,2,3,x,y\}$. There are $5!$ ways to rearrange the elements in the set, but I am confused about how to find the number of ways to rearrange the set given that $3$ comes ...
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3answers
50 views

How do I calculate the number of permutations of the list $(6, 6 ,5, 4)$?

I have the list $l = (6, 6, 5, 4)$ and want to how to calculate the possible number of permutations. By using brute force I know that there are 12 possible permutations: $$\{(6, 5, 6, 4), (6, 6, 5, ...
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1answer
144 views

$x^3=y^3=1, xyx=yxy$

The statement of the following problem from Artin's book is: Use the Todd-Coxeter algorithm to identify the number of elements in the group $G$ with the following defining relations: $x^3=y^3=1, ...
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3answers
53 views

Basic Permutation and Combinations practice quesiton

I am a novice at discrete mathematics and I have been working on trying to get my combinatorical skills up and i was working on some practice questions for permutation practice and i came across this ...
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1answer
334 views

Logic Pizza Toppings Ordering Question

So here's the question: The menu at a pizza place offers 14 possible toppings from 3 categories. Customers circle the toppings that they want on the order pad. (The order of circles or order of ...
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2answers
108 views

number of pairs formed from $2n$ people sitting in a circle

I am trying to understand the solution to the following problem: Suppose that $2n$ persons are sitting in a circle. In how many ways can they form $n$ pairs if no two adjacent persons can form a ...
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1answer
30 views

Probability : Container arrangement

There are a total of 15 containers out of which two containers have same color and the remaining are of different colors. The question is to find the probability that i) Two containers with same ...
1
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1answer
41 views

How should I continue my proof of this cycle property? (And did I make a mistake?)

I am trying to show: For a given single cycle, such as $(1, 4, 5, 7)$, the order of such a cycle is the length of the cycle. (i.e $(1, 4, 5, 7)^4 = ()$). I am trying to do this by induction. ...
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3answers
80 views

How to count permutations with cycles of length at least 51 in $S_{100}$?

Let consider permutation $ \in S_{100} $ How to count the number of permutations of those which contains a cycle of length 51 at least. ( so I would like a cycle of length 52,53,54,....,100)
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1answer
57 views

A simple question about permutations [closed]

So I could not find an answer anywhere, so here it is: If a string could be consisted of x y x y x y x y and x could only be used once, while y could be repeated, would it be correct to say that ...
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2answers
64 views

Combination and Permutation Math Problem

I am having some difficulty dissecting this problem and solving it: The track team has 7 girls and 6 boys. For the meet next week, they must choose a runner, a pole-vaulter, a captain and a ...
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1answer
49 views

Square labeled with same number.

Recently I met this combinatorics problem: "Let all points with integer coordinates in a plane be labeled with one of the numbers $1,2,3,...,n$. Prove that there is a rectangle whose vertices are ...
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2answers
262 views

How to maximize sum of pairwise multiplication of array elements taken one from each array?

Suppose you are given two arrays: $$a = [a_1,a_2,a_3,\dots,a_n],\hspace{5mm} b = [b_1,b_2,b_3,\dots,b_n]$$ Now you need to take one element from $a$ and one from $b$ multiply it, and add to sum and ...
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0answers
27 views

Form of an element of a normal subgroup of $A_n$

I want to show that $A_n$ is simple for $n\geq 5$. For $n=5$ I have used the following criterion Let $H$ be a normal subgroup of $A_5$ then $H$ can contain any one of the following $a.$ a $5$ ...
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0answers
130 views

Possible combinations of N different balls in M identical bins with the same capacity L

For a distribution center I am interested in the number of possible combinations to put N different boxes (all the same size but different content) into M equal bins (containers) with capacity L, with ...
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1answer
79 views

How many different permutations?

Suppose I've n boxes and m different colored balls of different quantities.How many unique permutations can be obtained ? Example : n=2,m=2, with quantities ( A - 1 ball, B - 2 balls) Thus the ...
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0answers
40 views

What is sgn(321)?

I've tried to compute the length of (321) and I got 2. Then the sgn should be (-1)^2=1. But I suppose sgn(321)=-1 by the definition in the graph?
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0answers
54 views

Please check my solution of a problem in combinatorics regarding partitions

A lift automatically operated has a further computer facility of recording how many people leave the lift at each floor. It starts at floor $1$ and goes up to floor $6$. If $8$ people consisting of ...
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1answer
100 views

Permutation question of arranging people in a row

There are $6$ boys and $4$ girls in a class. How many ways are there to arrange them in a row if no girl stands next to each other? I would know how to solve this if there are only $2$ girls. But ...
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1answer
41 views

Probability distribution of number of ordered items in a permutation

I have a simple algorithm to check if a series of numbers is sorted: if the first two numbers are sorted, move to the next two. Else, stop and return false. I want to figure out the average case ...
3
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1answer
94 views

To determine number of arrangements of 4 letters in a word so that the transitions remains conserved

A 10 letter word is composed of $A,\ B,\ C,\ D$. The problem is to find the number of arrangements of these alphabets which could lead to fixed number of transitions between each pair of alphabets. ...
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1answer
27 views

How do you do this permutation? [closed]

How would you do this math problem? and how do you know to use the equation you use? Mendy’s offers three types of bread: White, Whole-Wheat and Rye. The choices of meat are corned beef, pastrami, ...
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3answers
61 views

Subgroups of a permutation group

The permutation group $S_{4}$ is defined as the group of all possible permutations of [1234]. i) Find the number of subgroups of $S_{4}$ that have order 2. ii) A: { [1234], [2143], [3412], [4321] } ...
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3answers
144 views

prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$ [duplicate]

How can i prove that : $$\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$$ i tried to prove it : by $$\binom{2n}{n} = \frac{(2n)!}{n!n!} = \frac{2^n ...
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2answers
83 views

How to prove this equality for Stirling numbers?

How can I prove that the following formula is true for Stirling numbers of first kind. $$\sum_{k=1}^n(-1)^k\left[\begin{matrix} n\\k\end{matrix}\right] =0$$ Actually I want to prove that number of ...
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0answers
32 views

Counting with permutations and counting ignoring permutations

I am given this problem: This problem was given to me in my computer science class but it has to do with permutation and I want to understand it mathematically first. let $c(n)$ be the number of ...
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1answer
120 views

Four Letter-envelop problem

A secretary writes four letters and the corresponding addresses on envelopes. If he inserts the letters in the envelopes at random irrespective of the addresses, (i) find the probability that only one ...
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2answers
114 views

Number of visible elements in a permutation

The following problem occurred to me the other day, and I've played around with a bit but can't seem to find a good solution: Consider a permutation $\pi$ of $\{1, 2,\ldots ,n\}$. For every positive ...
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0answers
25 views

probability of vector in column span

Consider we have a fixed matrix M of size a$\times$ 2b (Let us look at M=[$M_1$ $M_2$] where matrices $M_1$,$M_2$ are of size a$\times $b) and a vector $v$ of dimension a. Is there any way that I can ...
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1answer
148 views

Outer automorphisms of the infinite symmetric group

Denote by S$_\infty$ the group of permutations of $\mathbb N$. Question: Does there exist an outer automorphism of S$_\infty$, and if so, can one be exhibited? Does this depend on the continuum ...
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1answer
57 views

Generators of symmetric group $S_n$ [duplicate]

How can you prove that $S_n$ is generated by $(1\space 2)$ and $(1\space 2\space 3\space ... \space n))$ for $n\geq 2$?
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1answer
16 views

Finding cycles with set of permutations

Let $\alpha = (\alpha_1 \, \alpha_2 \, \ldots \, \alpha_s)$ be a cycle, for positive integers $\alpha_1 , \alpha_2 , \ldots , \alpha_s$. Let $\pi$ be any permutation. Show that $\pi \alpha \pi^{-1}$ ...
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1answer
57 views

Does the product of all conjugates of some subgroup is independent of the order?

Let $G$ be a finite group and $A \le G$. Let $A^G = \{ A_1, A_2, \ldots, A_n \}$ be all the conjugates of $A$, i.e. each $A_i$ equals $A^g$ for some $g \in G$. Then I want to show that $$ A_1 A_2 ...
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1answer
32 views

permutations of 10 objects in a subset contains similar elements

A board that is divided into 15 different places, and we want to place 10 components on this board such that each component is placed in a section; knowing that those components are divided into 4 ...
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3answers
137 views

Simple Probability Question about Combinations

If someone could please point me in the right direction on these. I get lost on how to think about them. In a game there are four holes with values 0, 1, 2, and 4. You are given 6 balls to shoot into ...
0
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1answer
26 views

Double Check Probability for Permutation

I have to find the sample space and a few probabilities here and I am wondering about if I am going down the right track for these. If I am incorrect, then please point me in the right direction, but ...
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2answers
49 views

Question about permutations: How to show $\sigma(P)=(-1)^{\imath(\sigma)}P$?

A permutation of a finite set $X$ is any bijection from $X$ to $X$. We denote by $S(X)$ the set of all permutations of $X$. If $I_n:=\{1, \ldots, n\}$ we write $S_n$ instead of $S(I_n)$. Define ...
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2answers
146 views

Parity of permutation example

I know the definition of parity of permutation. But what does that look like in examples? For example, if the number of permutations is odd, then the sign of permutation in $-1$. What does this mean? ...