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

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1answer
37 views

How many ways of selecting from identical pairs?

My question is with regards to combinations and permutations. How many ways are there to select n unique objects from x number of identical object pairs? To make this question clearer, here is a ...
7
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3answers
155 views

Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
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1answer
66 views

Finding the 'n'th k-permutation of a set, and finding 'n' for a given k-permutation (lexicographical ordering)

Suppose you have a set, and want to order the k-permutations of the set (for example, the permutations of 5 elements of the set {1, 2, 3, ..., 16}). Is there a fast way of finding 'n' (the ...
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1answer
37 views

Permutations expressed as product of transpositions

There is a theorem that states that all permutations can be expressed as a product of transpositions. I have a couple of questions about this theorem: Does the product which is equal to the ...
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1answer
26 views

AM I doing this right? - How many binary words of length 8 are there that contain at least six 1's?

How many binary words of length 8 are there that contain at least six 1's? This is what I have: 8!/6!2! = 28 words Is this the correct answer?
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1answer
45 views

How to mathematically calculate the indistinguisable and distinct of the following permutation problems?

I'm having trouble calculating how many indistinguishable and distinct solutions there are for each problems. I'm pretty confident with some of my solutions, but could anyone show me mathematically ...
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2answers
57 views

Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$?

What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers. I've ...
2
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2answers
65 views

How do I prove that an order of a cycle is its length?

Let $\sigma$ be a cycle with length $n$ where $\sigma \in S_m$. How do i prove that $|\langle \sigma \rangle |$ is $n$?
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2answers
97 views

Finding Distinct Elements and Permutation in Partitioned Set

I am having a hard time figuring out where to start on a homework problem. The question is: A set of $nk$ elements is partitioned into $k$ subsets in two ways, each subset having size $n$: one ...
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3answers
224 views

A Problem of Combinatorics

In how many ways can three distinct numbers be chosen from the set {1,2,3,4....2n} such that the numbers are in increasing arithmetic progression?
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1answer
35 views

Combinations and Permutations. integer solutions [closed]

(a) How many integer solutions are there to the equation $x + y + z = 15$ if (i) $x$, $y$, $z$ are non-negative? (ii) $x$, $y$, $z$ are positive? (iii) $x$, $y$, $z$ are non-negative and $z \leq 5$? ...
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2answers
36 views

$(34)(123)(456)$ is a cycle. True or False?

I know this is basics, and I understand that $(34)(123)(456)$ is a product of cycles which, I found: $(124563)$. But somehow, I was lost. How do I know if it is indeed a cycle? OR if it isn't? Any ...
0
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1answer
21 views

permutations and combination

How many different strings of lights can be created by placing 40 coloured lights on a horizontal string if 12 of them are red, 6 are blue, 14 are green and 8 are yellow? Assume that lights of the same ...
0
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1answer
49 views

Bridge hand Combination/Permutation

A Bridge hand consists of 13 cards from a deck of 52 cards. In how many ways can a (bridge) hand consisting of 5 spades(♠), 4 hearts(♥), 4 diamonds(♦) and 0 clubs(♣) be selected?
2
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1answer
26 views

In how many ways can letters in mathematics be ordered with restrictions?

I've been stuck on these for a while. Please guide me through all the steps because I actually want to understand this. I've got an exam coming up. Consider the letters in the word "MATHEMATICS". In ...
2
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2answers
74 views

Algorithm to compute maximum permutation sum in matrix

Given a matrix $A_{n\times n}$ of real numbers, what fast algorithms do there exist to compute the maximum value of $a_{1,\sigma(1)}+a_{2,\sigma(2)}+\ldots+a_{n,\sigma(n)}$ over all permutations ...
0
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1answer
23 views

My proof regarding composition of permutations came to the same conclusion as the answer sheet, but through different methods. Is it valid?

Let $S_3$ be a set of all permutations of elements in $\{1,2,3\}$. Prove that there doesn't exist f $\in S_3$ where $\{f,f^2,f^3,f^4,f^5,f^6\} = S_3$. Where $f^n = f \circ f \circ \:... \circ \:f$ ...
2
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2answers
73 views

The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
0
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1answer
32 views

Do disjoint cycles commute?

When a given set is finite it is clear. I'm asking the general case. Let $X$ is an arbitrary set. Let $\sigma,\tau$ be disjoint cycles on $X$. Then do they commute?
0
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1answer
34 views

How many way can 7 friends line up if there are certain conditions?

How many ways can 7 friends line up if Ann, Beth, and Chris have to stand next to each other where Ann is ahead of Beth and Beth is ahead of Chris? Would it simply be $5*4*3*2*1=120$ ways? Expanding ...
0
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1answer
35 views

What is the definition of “disjoint cycles”?

I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics. Below, i list my definitions for cycle and orbit. Let $X$ be a nonempty set. Let ...
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2answers
41 views

What is “the orbit of a permutation”? Is the term “orbit” consistent with that for group action?

reference: What is the orbit of a permutation? To be honest, i don't understand the answer in the link. The orbit of a group action is defined as follows: Let $G$ be a group acting on a set $X$. ...
0
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1answer
37 views

Prove that $sgn(\sigma_1 \circ \sigma_2) = sgn(\sigma_1)sgn(\sigma_2)$

Lete $n\in \mathbb{N}$. Show that the transformation $$sgn: S_n \rightarrow \{\pm 1\}$$ (where $S_n$ is the set of all permutations of the integers in the set $\{1,...,n\}$),given by $\sigma \mapsto ...
0
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1answer
43 views

Analog of Birkhoff's theorem for doubly stochastic matrices

Birkhoff's theorem states that extreme point of the set of doubly stochastic matrices are permutation matrices. An $n \times n$ matrix $A$ is doubly stochastic if each row and column sums to 1. What ...
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2answers
35 views
1
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1answer
37 views

Permutations of two photo frames

Please help with this permutations question. I'm trying to use the permutation formula to calculate it but don't know where to begin: $$\frac{n!}{(n-r)!}$$ Here's the problem: ...
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2answers
25 views

permutation with four fixed numbers [closed]

My problem appeared to be part of permutation but not sure. I have a fixed length of 4 digits with 2 variable digits. say i have ...
1
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1answer
39 views

Envelopes and Mailboxes

We suppose $n$ and $p$ are two positive integers. A) In how many ways can you divide $p$ identical envelopes in $n$ mailboxes? (Each mailbox can hold several envelopes at the same time) B) In how ...
0
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1answer
26 views

Count no. Of ways

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.
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1answer
36 views

Probability of item distribution with a restriction

I'm having a hard time analyzing my research data, and was wondering if anyone had any suggestions? I've reworded the question so it is presented more like a statistics problem. There are $x$ number ...
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2answers
44 views

Sum of Binomial Series of form $\binom{2000}{3k-1}$

Find the Value of $$ \binom{2000}{2}+\binom{2000}{5}+\binom{2000}{8}+\cdots+\binom{2000}{1997}+\binom{2000}{2000}$$
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2answers
155 views

(12345) is an even permutation of S_5. True or False?

The answer i had for this question was True, yet i'm not sure. Well, from what I know so far was that: $(12345)$ can be expressed as a number of 4 transpositions such as: $(12)(23)(34)(45)$ which is ...
0
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1answer
42 views

Conjugate subgroups of $S_4$

$A = \langle (1,2,3),(1,2)\rangle$ $B = \langle (1,2,4),(1,2)\rangle$ $C = \langle (1,3,4),(1,3)\rangle$ $D = \langle (2,3,4),(2,3)\rangle$ I want to proof that these subgroups of $S_4$ ( which ...
2
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3answers
245 views

How do I solve for n in this permutation question?

I have the following question: Solve for n: $$_nP_3 = 6_{n-1}P_2$$ I don't know how I should begin to tackle this problem? Any tips/help would be appreciated.
2
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2answers
45 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
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0answers
26 views

Fixed points and permutations.

Let $\psi ,\varphi \in {S_n}$ two permutations. Let $M$ a matrix such that $a_{i,j}=1$ iff $i=\sigma(j)$ where $\sigma \in S_n$ ($0$, otherwise) I already showed that $tr(M) = \left| {\left\{ {k \in ...
3
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0answers
25 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
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2answers
125 views

Number of ways to sit 6 girls and 6 boys together with no two girls together.

As the title of the question explains: What I thought on the very first instant was that we will make them sit alternate hence the answer will be 2 * 6! * 6! But ...
0
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1answer
40 views

four digit numbers that have at least one of their digits repeated

The number of four digit telephone numbers that have at least one of their digits repeated is 9000 4464 4000 3986
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1answer
36 views

Finding the maximum possible order for an element in $S_5$

I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order. But my question is, do I have to write all elements of $S_5$, ...
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1answer
44 views

How many distinct elements does a group of permutation on 3 letters have?

I am having some problems solving a problem similar to this. So i tried making it a more simpler problem. I really don't know how to approach this kind of problem. A hint would be very much ...
0
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3answers
62 views

Forming 4-digit odd numbers under certain rules [closed]

How many four-digit odd numbers can be formed such that every $"3"$ in the number is followed by a $"6"$? A) 108 B) 2592 C) 2696 D) 2700
0
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2answers
46 views

finding the password [closed]

Charlie has forgotten his six-digit ID number. he remembers the following: the first two digits are either 1,5 or 2,6, the number is even and 6 appears twice. if raju uses a trial and error process to ...
0
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0answers
33 views

why are these cosets equal?

Please disregard this question until I have uploaded a screenshot K is the subgroup of S_3 defined by the permutations {(1), (123), (132)} They have (1)K = (12)K = {(1), (12)} What they did was ...
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2answers
56 views

Coloring vertices of a square

Using four colors, red, white, blue and green, in how many ways can the vertices of a square be colored? Assume that reflections and rotations are allowed, meaning that if you examine a square from ...
0
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1answer
70 views

Number of seating arrangements in 5 cars

An exercise from Introductory Combinatorics by Richard A.Brualdi: A roller coaster has five cars, each containing four seats, two in front and two in back. There are 20 people ready for a ride. ...
0
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1answer
16 views

number of elements in unsortet case

I have a group M with Mn different elements. How many unique combinations can I make out of this in an n digit system when order is no importance. For example if M = {1 2} & n = 3 ...
1
vote
1answer
57 views

How many ways are there to place these books on the shelves?

You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.) I want to say $7^5$ since there are 7 possible shelves and ...
1
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1answer
28 views

triangles and lines

There are 12 points in a plane. If 4 of them are on a straight line and no other 3 points are on a straight line, then find the difference between the number of triangles and the number of straight ...
0
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1answer
23 views

arrangement of balls in bowls

There are five bowls numbered $1$ to $5$. There are $5$ green balls and $6$ black balls. Each bowl is to be filled by either a green or black ball and no two adjacent bowls can be filled by green ...