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

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Permutations of n beads on a string.

Suppose you have n beads on a string, the beads are labeled 1,2,3,...,n. How many possible permutations are there if you are allowed to flip the string over? E.g. For 4 beads, 1--2--3--4 is ...
2
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3answers
145 views

What is the number of combinations of the solutions to $a+b+c=7$ in $\mathbb{N}$?

My professor gave me this problem: Find the number of combinations of the integer solutions to the equation $a+b+c=7$ using combinatorics. Thank you. UPDATE Positive solutions
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1answer
92 views

Permutations and combinations two part question

This is for some personal combinations / permutations study, I was wondering about a certain type of question that I shall phrase thus: Given 17 of object A and 13 of object B, how many ways may four ...
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3answers
434 views

Summation of a finite series involving permutations.

$$\large \sum_{i = 2}^{25}P(i,2)$$ $P$ stands for "permutations".
3
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3answers
397 views

How to get the N-th word in a sequence

Suppose I have an alphabet (e.g. consisting of ABCDEF) and a lexicographic order is defined i.e. A -> B ... -> F -> AA -> AB .. -> AF -> BA -> BB -> ... -> BF ... -> FF -> AAA -> ... Is there a ...
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2answers
154 views

confusion over transpositions and cycle notation

Suppose we have a permutation on the set {1,2,3,4,5} and we express it in the cycle notation as (2,5,3). I interpret this to mean that every time we apply the permutation, 2 gets sent to 5, 5 gets ...
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0answers
180 views

Who invented the breadth-first permutation algorithm?

My initial problem was solved here. It is about enumerating all n-tuples of a permutation in a specific order. The solution algorithm is very simple and I'm sure has been used before. However, I did ...
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1answer
878 views

GRE Permutation and geometry question

if you join all the vertices of a heptagon, how many quadrilaterals will you get? How many heptagons can be drawn by joining the vertices of a polygon with 10 sides? A polygon has 20 diagonals,how ...
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2answers
1k views

How many paths possible in this grid given this specific conditions?

In a grid, There is a bottom-left point i, with co-ordinate (0,0) and top-right point j with co-ordinate (10,10). A person is standing at point i, He can go up (1 unit), right (1 unit) and diagonally ...
0
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1answer
165 views

Number of arrangements of 2 objects which contains one of them at at least “m” contiguous places. [duplicate]

Possible Duplicate: Counting subsets containing three consecutive elements (previously Summation over large values of nCr) Suppose we have large number of two types of objects $A$ and $B$. ...
1
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1answer
515 views

how many ways we can choose 3 and more consecutive number from set of N numbers [duplicate]

Possible Duplicate: Counting subsets containing three consecutive elements (previously Summation over large values of nCr) Suppose we have a set like (1,2,3) then there is only one way to ...
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0answers
83 views

Perfoming permutation count

I have a question, probably relatively simple one. Once you want to find all the unique combinations of number 5 you count 5! = 120. Once you want to find all the combinations including repetition of ...
23
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1answer
499 views

“Efficient version” of Cayley's Theorem in Group Theory

I'm considering finite groups only. Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$. I think it's interesting to ask for smaller values of $n$ for which $G$ is a ...
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2answers
368 views

How does a permutation of roots induce an automorphism on a splitting field?

I'll be learning Galois theory for the first time later this year and wanted to clear up something that was puzzling me. If f(x) = $\sum {a_ix^{i}}$ is a polynomial which is irreducible over ...
2
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1answer
474 views

About circular permutation

I know this is a physics group, but then I think you guys can answer me .. anyone can explain how circular permutation work? From the explanation from the book, it is understood that if CW $\ne$ ACW , ...
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1answer
372 views

Given algorithm which prints $n$-th string of nested parentheses, find a reverse algorithm

We have the following balanced brackets permutations of length $4\cdot 2$ in lexicographical order: ...
4
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1answer
384 views

$N^\text{th}$ (in lexicographical order) term of balanced brackets string

We have the following balanced brackets permutations of length $4\cdot2$ in lexicographical order: ...
2
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1answer
81 views

Permutation across matrices.

Matrices may be used to permute the order of elements in a set. For example: $$ \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 ...
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2answers
997 views

How to calculate the expected frequency of a pattern?

I'm working on a problem to find the expected frequency of a pattern. Say there is a sequence of alphabets - A, B, C and D. The sequence is: ABDACDBADA. I want to find the expected frequency of a ...
0
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2answers
196 views

In how many ways a number $\gt 5000$ can be formed using given digits without repeating?

In how many ways one can form a number greater than $5000$ when allowed only to arrange digits taken from $2,3,4,5,8$ without repeating any digit? I would think it would be $2\cdot4\cdot3\cdot2$. ...
2
votes
1answer
391 views

Counting necklaces with a fixed number of each bead

I want to count the number of necklaces, with $n$ beads in total, where the alphabet of beads is $\{1,\ldots,k\}$, and where the number of beads with color $i$ is $n_i$. For example, if $n=4$, and ...
5
votes
1answer
120 views

Partial recurrence relation for the number of permutations in $S_n$ which have a square root.

I ran into this problem the other day. The proof is supposed to be done by exhibiting an explicit bijection between two sets, without using induction, recurrence, or generating functions. Denote ...
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4answers
1k views

Ways of selecting at least one man

We have to select a group of 7 out of a group of 9 men and 11 women Q : How many seven member teams consist of at least one man ? Now I know that the answer is ${20 \choose 7}-{11 \choose 7} ...
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1answer
145 views

Integer Partition by Counting Repetition : Conjecture ??

I would like to find informations regarding this way of doing Integer Partitions or this conjecture, Suppose you have all the ordered partitions of 5: 5 4 1 3 2 2 2 1 3 1 1 2 1 1 1 1 1 1 1 1 Then ...
2
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1answer
115 views

Is my solution to this permutation question correct?

Question: Suppose there are m girls and n boys in a class. What is the number of ways of arranging them in a line so that all the girls are together? (Biggs, Discrete Mathematics 2nd ed, Exercise ...
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1answer
1k views

Dealing with “at least” in Permutation

For the following question (which I pulled of the internet) A five member committee is to be selected from among four Math teachers and five English teachers. In how many different ways can ...
3
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2answers
505 views

odds of picking exactly 2 women and 2 men out of 12 men and 12 womem

I understand the answer to be 12 choose 2 * 12 choose 2 over 24 choose 4. I don't really understand why, or what principle I can extract from the problem. I can understand that we are putting the ...
0
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2answers
105 views

Help on groups and symmetries homomorphisms

I need help on question 1 I am preparing for my test: 1) Find a homomorphism $\alpha: A_4 \to\mathbb Z_6$ such that $\ker (\alpha) = K$ where $K$ is the normal subgroup $K= \{(1), (12)(34), ...
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votes
2answers
419 views

How many 4 worded sentences can a list of 5 words make if two of them must be in that sentence?

Suppose we have: I am new at this - (5 words) how many 4 worded sentences can we make with this if "new" and "this" must appear in the sentence. I think its : .# of sentences we can make with any ...
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1answer
111 views

Finding the number of distinct elements in a group generated by a permutation and a function over tuple.

Okay, I'm not sure if the title is correct but is there any general way of finding the number of distinct elements that are generated by some function (for example $*$) and a permutation (in this case ...
4
votes
1answer
415 views

Permutations of a set with a conditional subset

Using the digits 1, 2, 3, 5, 6, 8, 0 only once, how many 4-digit numbers could be constructed if the number is even? This is an exercise from an online course I'm taking. The given solution suggests ...
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3answers
6k views

Distinct permutations of the word “toffee”

What does distinct permutations mean and how many distinct permutations can be formed from all the letters of word TOFFEE?
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0answers
299 views

sum with permutations

Let $a$ be vector in $R^{2m}$. And let $S_{2m}$ be group of all permutations on the set $\{1,\dots,2m\}$. I would like to calculate $$ \sup_{\pi\in S_{2m}}\sum_{d(\sigma, ...
0
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1answer
94 views

permutation within a combination

I am having a bit of a problem with the following question: There are 16 balls, 5 red, 8 blue, and 3 green. The result is the ordered list of the first 4 balls only (although all balls have been ...
8
votes
1answer
192 views

Subgroups between $S_n$ and $S_{n+1}$

Lets look at $S_n$ as subgroup of $S_{n+1}$. How many subgroups $H$, $S_{n} \subseteq H \subseteq S_{n+1}$ there are ?
4
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1answer
283 views

question involving Markov chain

Let $S_{2m}$ be the group of all permutations $\pi$ of $\{1, 2, \ldots, 2m\}$. The following transition kernel $S$ generates the random transposition walk $$ Ch(\pi, \pi')= \begin{cases} \frac{1}{2m} ...
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2answers
271 views

Permutation problem scheduling games

There are: 14 teams $t$ numbered from 0 to 13 13 different games $g$ numbered from 0 to 12 13 rounds $r$ numbered from 0 to 12 I want to make a planning such that: each team plays against ...
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2answers
53 views

Combinatorics: how many unique albums…

I want to record a set of music albums so that each one is unique. Each album has ten tracks, and I've recorded 4 versions of each track. How many unique albums can I compile so that no two albums has ...
2
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2answers
97 views

Bijection from $S_{n-1}$ to $\{\sigma \in S_{n} : \sigma(k) = j \}$

Let $n$ be a natural number. Let $k$ be an element of $\{1, \ldots , n\}$. For each j in $\{1, \ldots , n\}$, I want to find a bijection $f_j$ from $S_{n-1}$ to $\{\sigma \in S_n : \sigma(k) = j ...
0
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2answers
83 views

About $|\operatorname{Sym}(\Omega)|$ when $\Omega$ is an infinite set.

Here is a problem: Show that if $\Omega$ is an infinite set, then $|\operatorname{Sym}(\Omega)|=2^{|\Omega|}$. I have worked on a problem related to a group that is $S=\bigcup_{n=1}^{\infty } ...
2
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6answers
114 views

Permutation seating arrangment 5!.3!

What wold be the answer for this How many ways can 3 boys and 4 girls sit in a row if all the boys are sit together. Answer listed as $5!\cdot3$! $4+3 = 7$ what is 5 doing here? someone please ...
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2answers
2k views

How to solve this permutation math problem?

In how many ways can 4 girls and 2 boys sit at a movie theater row with 6 seats if a girl must be seated at each end.
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4answers
204 views

Selection using permutation and combination

From 4 men and 4 ladies a committee of 5 is to be formed. The committee consists of a president, vice president and three secretaries. What will be the number of ways of selecting the ...
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4answers
7k views

Difference between permutation and combination?

Permutation: $$P(n,r) = \frac{n!}{(n-r)!}$$ Combination: $$C(n,r) = \frac{n!}{(n-r)!r!}$$ Apparently, you use combination when the order doesn't matter. Great. I see how a combination will give you ...
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2answers
3k views

De-arrangement in permutation and combination [duplicate]

This article talks about de-arrangement in permutation combination. Funda 1: De-arrangement If $n$ distinct items are arranged in a row, then the number of ways they can be rearranged such ...
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2answers
563 views

normal subgroups of infinite symmetric group

I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the ...
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1answer
169 views

Counting in how many ways rocks can be put in boxes

How can I figure out the following questions? How many possible combinations can be done by having 26 boxes and 15 red rocks, and 15 black rocks? Each box can have up to 15 rocks in it. We can have ...
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1answer
72 views

Finding an imprimitive group on $12$ letters

By definition a permutation group $G$ acting on a set $\Omega$ is called primitive if $G$ acts transitively on $\Omega$ and $G$ preserves no nontrivial blocks of $\Omega$. Otherwise, if the group does ...
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1answer
136 views

Fixed Block is an orbit?

Reviewing some of my old questions here, I am stuck at a comment in which Prof. Holt gave me an interesting example (A small one) about non-transitive $1/2-$transitive group. Here is the link ...
3
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1answer
103 views

Need help about $P\Gamma L_2(q)$, $q=4,3$

I am asking kindly, For which values of $n$ we have $$S_n≅P\Gamma L_2(3),S_n≅P\Gamma L_2(4)$$ This may be correct if we replace $S_n$ by $A_n$. Any help will be appreciated. :) Edit (JL): ...