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

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If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
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0answers
76 views

The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
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1answer
18 views

No. of ways to shuffle a card

How many ways are there to shuffle N cards such that exactly one card is in the same position?(Assuming that initially the card no. 1 is in the first position,card no.2 is in the second position and ...
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1answer
150 views

Product of permutations of consecutive numbers yields arithmetic sequence

Let $n\geq 3$ be an integer, and $a,b$ be positive integers. Let $c_1,\ldots,c_n$ be a permutation of $a,a+1,\ldots,a+(n-1)$, and $d_1,\ldots,d_n$ be a permutation of $b,b+1,\ldots,b+(n-1)$. Is it ...
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1answer
26 views

Every solution of a permutation equation?

I have the three permutations $$a=(1\;3\;4\;8),\quad b=(2\;3\;5\;7),\quad c=(4\;3\;2\;8)$$ and I have to find all $x$ satisfying $$axb=c.$$ I have found one solution (I hope it's good): ...
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5answers
82 views

There are $n$ persons sitting around a table…

There are $n$ persons sitting around a circular table. Then, in how many different ways 3 persons can be selected if none of them are neighbours. My approach:- Let us pretend that we have already ...
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3answers
33 views

Finding number of integral solutions

I am really getting confused in this question. Number of integral solutions of the equation. $x_1x_2x_3x_4=770$ options- $2^{11}$ $2^{10}$ $4^4$ $5^5$ I attemtemted it by saying that ...
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3answers
65 views

Why is this combinatoric solution not correct?

I'm trying to solve the following problem: Balls of the colors red, orange, yellow, green, blue, indigo, violet (7 colors, 1 ball per color) are placed into 4 different boxes A,B,C,D so that no box ...
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2answers
104 views

Christmas protocol

Since holiday season is coming, here is a little practical-purpose combinatorics question. Lots of group of friends or families practice the random variant of Secret Santa, where each member buys a ...
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2answers
71 views

Combinatorics in a Party.

There are 12 persons in a dinner party, they are to be arranged on two sides of a rectangular table. Supposing that the master and the mistress of the house have are always facing each other, and ...
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3answers
65 views

Why is this $6!$ factorial and not $p(6,1)$?

There is this question. There are six different candidates for governor of a state. In how many different orders can the names if the candidates be printed on a ballot? The answer is $6!=720$. But ...
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2answers
51 views

Forming a committee

Suppose a committee must be formed from a group of 15 professors and 10 administrators. How many committees can be formed if the committee must consist of 5 professors and 5 administrators? Update 1: ...
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1answer
46 views

There are 10 stations on a circular path

There are 10 stations on a circular path. A train has to stop at 4 stations such that no two stations are adjacent. How many such selections are there?? Since i know if the stations are on the ...
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1answer
31 views

Simplifying Permutations

Could someone explain the process of simplifying the following permutation in $S_6$ (1,3,5)(2,4,5)(2,3,6) An explanation on how you arrived at the simplified form would also be greatly appreciated. ...
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2answers
33 views

How many ways are there to place 7 distinct balls into 3 distinct boxes?

How many ways are there to place $7$ distinct balls into $3$ distinct boxes? is the question I'm confused about. The solution shows that the correct answer is $3^7$. I'm just confused why this is. ...
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4answers
439 views

Is it always true that $(a,b,c)(a,b,c) = (a,c,b)$?

I noticed that $(1,2,3)(1,2,3) = (1,3,2)$, and I also noticed that $(1,4,3)(1,4,3) = (1,3,4)$. Now, my question is whether or not it is true that for any permutation $(a,b,c)^2 = (a,c,b)$?
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1answer
28 views

Number of permutations of m objects taken out of n objects where an object can repeat any number of times.

I'm given $n$ distinct objects. In how many ways can we select and permute $m$ objects out of those $n$ objects. $n$ may be less than $m$ and any object can appear any number of times. For example: ...
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2answers
24 views

License plate permutations [closed]

If a license plate is to have 3 letters followed by 3 digits and repetition of letters and digits is allowed how many different license plates can be made?
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1answer
32 views

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular.

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular. I proved that if $N$ is intransitive, then $G$ will be imprimitive and Frobenius but I don't ...
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1answer
73 views

The number of nonnegative integer triples with sum equal to $12$

How many triples $x,y,z$ satisfy $x+y+z=12$, where $x,y,z\ge 0$ are integers? Note that these are not ordered: $ (12,0,0)$ and $(0,12,0)$ are treated as same. Progress I have tried the formula ...
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0answers
48 views

Other Patterns in Triples

I have the following 20 triples generated by polynomial distribution: $$\begin{matrix} (2,4,5)&(2,3,4)&(2,3,5)&(1,4,5)&(2,2,4)\\ ...
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2answers
42 views

In how many ways can I merge $m$ and $n$ items without disturbing the order in each group?

I have two lists having all distinct elements. One contains $m$ elements and other contains $n$ elements. We need to arrange them such that the order of elements of individual lists is not disturbed. ...
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1answer
33 views

How many ways are possible to place k items in n spots such that order of k items is not disturbed

I have k items, need to place them in n spots(n>k). In how many ways can this be done? Example - for k=2 and n=4, these are the possibilities assuming items to be like this [1,2] 12-- 1-2- 1--2 -12- ...
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0answers
20 views

What is the algebra behind the Cuthill Mckee Bandwidth Reduction

From my understanding a Sparse matrix is converted to banded matrix and then the cuthill mckee is used to reduce the bandwidth. I have spent about three days browsing the web to find an example where ...
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3answers
43 views

No. of different possible arrangements.

How can I find no. of different possible arrangements with the factor of the term $a^2b^4c^5$ written at full length.
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3answers
56 views

if $m, n \in \mathbb{N}$, $m < n$, then $S_m$ isomorphic to subgroup of $S_n$

How do show that if $m, n \in \mathbb{N}$ and $m < n$, then $S_m$ is isomorphic to a subgroup of $S_n$, without using any "overpowered" results?
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3answers
20 views

Showing product of disjoint cycle

I am trying to show the product of two disjoint cycles such that they have nothing in common for $A_n$ for $n\ge 3$. So I have the two cycles $(ab)(cd)$. I have read here: ...
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3answers
45 views

Probability question - arranging 20 pupils in a row - 8 boys and 12 girls

We have 20 pupils in class, 12 girls and 8 boys. We arrange the pupils in a row, and now need to calculate the following probability: a. The probability that Jana, one of the girls, will not stand ...
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0answers
61 views

Approach name - Ross Millikan's answer

I want to know the name of an approach (formula) in the first comment of this question (@Ross Millikan's answer) Counting arrays with gcd 1 Thanks
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1answer
100 views

Select k no.s from 1 to N with replacement to have a set with at least one co-prime pair

Given $1$ to $N$ numbers. You have to make array of $k$ no.s using those no.s, where repetition of same no. is also allowed, such that at least one pair in that chosen array is co-prime. Find no. of ...
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48 views

Permutation order

How do you put non disjoint permutation cycles into disjoint cycle form? For Example the permutation in non disjoint cycle form (1352)(34)? How do you form disjoint cycle for from this?
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Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
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2answers
57 views

Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive?

Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
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Grandfather-Grandchildren Family Photograph Combinatorics Problem.

In how many different ways a grandfather along with two of his grandsons and four granddaughters can be seated in a line for a photograph so that he is always in the middle and the two grandsons are ...
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1answer
13 views

If I'm counting the number of binary strings of a certain length with a certain number of 1's, should I use combinations or permutations?

And should I use repetition allowed, or repetition not allowed formula? A binary string is a string with 1's and 0's in a row. {0,1} is a different string from {1,0}. Say I'm considering binary ...
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5answers
270 views

Permute “aaaaabbbbbccccc” so that no two identical letters are adjacent

This is a follow up question to Application of PIE. How many strings with the letters "aaaaabbbbbccccc" are there so that no two identical letters are adjacent?
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1answer
53 views

Let there be 9 fixed point on the circumference of a circle.

Let there be 9 fixed points on the circumference of a circle. Each of these points is joined to every one of the remaining 8 points by a straight line and the points are positioned on the ...
2
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1answer
55 views

How many ways are there to arrange k out of n elements in a circle with repetition?

If you a set of the n elements, in how many ways $Q(n,k)$ can you take some of them and arrange them on a $k$-gon, when repetition of one element is allowed but rotations of one arrangement are not ...
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3answers
58 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
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Distance between element and its position in permutation

Consider a permutation $\pi$ that is chosen uniformly at random among all permutations of $\{1, \dotsc, n \}$. Let $a_i$ be the position of $i$. We want to find $$E[\sum_{i=1}^n |a_i -i |]= ...
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0answers
30 views

Need a hint with permutations and pigeonhole-principle question

let $\pi_1,\pi_2,\pi_3\in S_{28}$. Help me prove that there are two sub-sequences of 28 with length 4 $i_1< i_2 <i_3<i_4,\ and\ \ j_1<j_2<j_3<j_4$ so that $\pi_q(i_n)=\pi_p(j_n)$ ...
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1answer
39 views

Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
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A test contains 10 T/F questions, 5 must be marked true, and 5 false…

A quiz consists of ten true/false questions. a) In how many distinct ways can the quiz be completed if no answers are left blank? b) In how many ways can the quiz be completed if five questions must ...
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25 views

How many different vertical arrangements are there of 10 flags if…?

How many different vertical arrangements are there of 10 flags if 4 are white, 3 are blue, 2 are green and 1 is red? I know the answer is 12 600 but am not sure how to get to it. Could someone walk ...
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0answers
11 views

Calculating Permutations of multiple, non-equal shapes on 2-Dimensional grid

I originally posted this in StackOverflow before finding this Mathematics forum. This is primarily a mathematics question so I am reposting here: Situation: The following components exist for this ...
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0answers
55 views

Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
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1answer
24 views

A circle and six sectors

A circle is divided into six sectors and six numbers 1,0,1,0,0,0 are written clockwise,one in each sector. in one step, we can add one to the numbers in any two adjacent sectors. Is it possible to ...
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1answer
36 views

Lagrange Theorem: left and right cosets

Thank you for taking the time to read this and help me out. This was a question I missed on my past test: "Let $G = S_3$ and $H = A_3$ ($H \subseteq G$ is a subgroup). Compute the left and right ...
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1answer
29 views

Seven people are interviewed for a possible promotion. In how many orders can the seven candidates be interviewed?

Seven people are interviewed for a possible promotion. In how many orders can the seven candidates be interviewed? I know the answer is 5040 but don't know how to get to it.
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PIN number consists of four letters, how many different PINs are possible?

The personal identification number (PIN) used by a certain automatic teller machine (ATM) is a sequence of four letters. a) How many different PINs are possible? Write the answer in exponential ...