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

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2
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1answer
62 views

different path permutation problem

Im having hard time with this question How many different paths in the $xy$-plane are there from $(0,0)$ to $(7,7)$ if a path proceeds either one space to the right $(r)$ or on space upward $(u)$? ...
2
votes
0answers
52 views

Prove there is $\sigma\in S_3$ such that $H_{\sigma (i)} \cong\textrm{}K_i ,\space \forall i $

In class they gave us a problem, After spending a long time trying to solve it, I turn to you =] Let $H_1,H_2,H_3, K_1,K_2,K_3 \le G$ be simple groups, $G=\{{h_1}{h_2}{h_3}:h_1\in H_1,h_2\in ...
0
votes
1answer
39 views

Round table combinatorics?

Sorry if this is a horrible format to read(first time using this site!). Your friends A, B, C and D are going to sit right next to eachother around a round table at your birthdayparty. You do not ...
-4
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0answers
50 views

Answer of a question

http://www.iarcs.org.in/inoi/2013/zio2013/zio2013-qpaper.pdf The fourth question. Can anyone explain how to solve it? EDIT: Sorry I didn't know about that. I need to calculate the number of ...
1
vote
1answer
18 views

Combination problem for N items in M identical groups

As the title says, I have N different items that will be put in M identical group, N >= M. The size of each group is not defined and the order of groups is not important. For example, if N = 4 and M ...
0
votes
2answers
50 views

sum of all the numbers that can be formed using the digits 2,3,3,4,4,4..

How to find the sum of all the numbers that can be formed using the digits 2,3,3,4,4,4.. What should be the way of doing this type of problem?? Please guide
0
votes
0answers
36 views

Given $k$ and $a$, find $n$ for $\frac{n!}{(n-k)!}=a$

Given the number of permutations of size $k$ for $n$ distinct objects $a$. Find the number of distinct objects $n$ (where $k \in \mathbb{Z}, 1 \leq k \leq n$) ? Basically, i'm given $k$ and $a$. I've ...
1
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2answers
36 views

Splitting of conjugacy class in $A_n$

During reading, I have encountered this, in several places: The following are equivalent for a permutation $\sigma \in A_n$: 1) the $S_n$-conjugacy class of $\sigma$ splits into two $A_n$-classes 2) ...
0
votes
1answer
30 views

How many sequences of five bases are there?

A and G are purines and C and T are pyrimidines. How many sequences of five bases are there that consist of three purines and two pyrimidines? I thought that I could do : 2^3 x 2^2 = 32 my teacher ...
1
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1answer
37 views

Doubt : Prove the number of matches

I was working my way through some problems in Discrete Maths by Rosen, when I came across the following question: There are x players in a singles badminton tournament Show that there are ...
5
votes
0answers
37 views

Number of representatives from states to from a comittee?

Among the three representatives to a conference from each of the fifty states, either none, one, or two of the representatives will be chosen for a large special committee. How many ways can this ...
1
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2answers
30 views

How many ways can 5 students be seated in a row of 6 chairs if 2 students must sit togegher?

The 2 adjacent students will have an arrangement of 5. Next, the open 4 chairs can be filled by either of the 3 students for 3!= 6. The product of 5 and 6 = 30. Am I taking the proper approach?
1
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1answer
30 views

Is there a closed form for this sequence?

I'm trying to find a closed form for the following sequence: $a$ $a(a-1)$ $a(a-1)(a-2)$ $a(a-1)(a-2)(a-3)$ The problem is, $a=\frac{1}{2}$. If it were some whole number, then I'd use ...
0
votes
1answer
23 views

lines in a plane.

Let there be 'p' lines which are concurrent at a point A and 'q' other are concurrent at a point B. Oh, and no 2 lines are parallel. Case 1: No line out of 'p' passes through B and similarly for 'q' ...
1
vote
1answer
36 views

What is the link between the kernel of a map and the map being 1-1?

When showing isomorphisms, I have sometimes (most likely incorrectly in cases) shown that, for a map $\theta : G \to H$, that $\operatorname{Ker}(\theta) = K \leq H$, means it is $1-1$. What is the ...
0
votes
1answer
27 views

Ways of writing an $n$-cycle as product of a $2$-cycle and $n-1$ cycle.

We know that any $n$ cycle can be written as a product of a $2$-cycle and an $n-1$ cycle; but this decomposition is not unique: $(123)=(12)(23)$ and $(123)=(23)(31)$ [product taken from right to left ...
0
votes
1answer
20 views

How many permutations of size 5 does the 4 make with 1st five numbers .repetition allowed.

What i want to calculate is suppose if we are having m numbers then how many permutations of size n will be there such that k fixed numbers are always present in those permutations. Example : we ...
1
vote
1answer
59 views

Coloring a Tree

We numberd nodes from $1$ to $N$ for convinent. We firstly color node $1$. Then we will color $N - 1$ remaining nodes, in any order which satisfied condition: node are chosen to color must be ...
0
votes
1answer
22 views

The commutator of any two transpositions in $S_n$.

Given any two transpositions $(i,j),(k,l) \in S_n$ such that $i,j,k,l$ are all distinct, is it true that: $[(i,j),(k,l)]=(i,j)(k,l)(i,j)(k,l)=[(i,j)(k,l)][(i,j)(k,l)]=\text{identity}$? I would ...
2
votes
3answers
43 views

A Question on circular permutations.

20 persons are to be seated around a circular table. Out of these 20 , 2 of them are brothers then number of arrangements in which there will be at least three persons between the brothers is.? SO ...
0
votes
1answer
19 views

Find the probability for … [duplicate]

Suppose we uniformly and randomly select permutations from the 20! Permutations of 1, 2, 3,..., 20. What is the probability that 2 appears at an earlier position than any other even number in ...
1
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1answer
70 views

Total no of permutations

How do i solve this type of problem: Suppose i have two sets $s_1 = \{a,b,c\}$ and $s_2 =\{ d,e\}$ now i have $5$ blocks which can be filled using any of theses $\{a,b,c,d,e,f,g,h\}$ Now the ...
1
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1answer
31 views

Number of permutations with repetition

I have to form a permutation of N elements with each element less than or equal to M. I need to count the number of unique permutations possible with 2 of the elements fixed as 2 and 3. Note that ...
1
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1answer
42 views

Number of lists of given size with given max element value where K cells (possibly overlapping) can have only multiples of certain numbers

I have been trying to wrap my head around a problem. The problem is reduced form of a problem from a programming contest. I have been trying to apply inclusion-exclusion principle toward solving this, ...
1
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0answers
23 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
1
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1answer
30 views

Product of $2$ permutations

$(2,3)(4,6,5,1,2)=?$ The multiplication is from right to left. I don't know, where I make the mistake. Denote $\tau=(2,3), \sigma=(4,6,5,1,2)$ $1\ \ 2\ \ 3\ \ 4\ \ 5\ \ 6$$\quad$ first apply ...
0
votes
1answer
30 views

Permutation inverse form

Given: $A=\{1,2,3,4,5,6\}$, $P_1=\begin{pmatrix} 1 &2& 3& 4& 5& 6\\ 2& 3& 4& 1& 5& 6\end{pmatrix}$, $P_2=\begin{pmatrix}1 &2 &3 &4& 5 ...
2
votes
2answers
83 views

Solving Rubik's cube and other permutation puzzles

I've seen two questions on solving the Rubik's cube but none of the answers have given a complete solution using mainly mathematical techniques. Furthermore, I've not seen a good explanation of ...
5
votes
1answer
57 views

Subgroups of $S_n$ that can send any subset of $[n]$ to any equally sized subset of $[n]$

This is a repost of a question I was trying to solve yesterday that got deleted. The question asked for a characterization of the subgroups $G$ of $S_n$ which when endowed with their natural action on ...
1
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3answers
35 views

Permutation in discrete math

Is the permutation $$\begin{pmatrix} 1& 2 &3 &4 &5 &6&7 \\ 7 & 4 & 2 & 1 & 3 & 6 & 5 \end{pmatrix}$$ even or odd? The product of disjoint cycles is ...
5
votes
5answers
781 views

How many Arrangement of “AMAZED” letter E Positioned between two A's (Not necessarily Flanked)

I considered 'AEA' as one letter so there are 4 letters which can be arranged in 4!=24 ways. But my sheet is telling its 120 How? Please HELP! & What is flanked meaning here?
0
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1answer
20 views

Maximal Permutations of numbers with monotonic objective function

I felt confident in the validity of the following statement, but now that I've played with the proof more I'm starting to have a few minor doubts. Any thoughts? Suppose you have two partitions of ...
1
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1answer
31 views

Write a transposition as a product of adjacent transpositions

I just read that any transposition can be written as a product of adjacent transpositions. I thought that I knew the right proof of this, but then I read that $\tau_{i,j} = \tau_{i,i-1} \circ...\circ ...
1
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0answers
32 views

Circular arrangement problem

I have one question of circular round table arrangement: " How to find the number of ways in which 6 persons out of 5 men and 5 women can be seated at around table such that 2 men are never together. ...
2
votes
1answer
29 views

Derangement, example, paradox?

How to explain that $!0 =1 $ and $!1=0$ I understand why $0! =1$. But when it comes to permutation $1!$ is also $1$. And in result it doesn't argue with my intuition. But, when it comes to ...
0
votes
2answers
65 views

Please explain definition of determinant using permutations?

Many people (in different texts) use the following famous definition of the determinant of a matrix $A$: \begin{align*} \det(A) = \sum_{\tau \in ...
2
votes
1answer
25 views

Permutations and combinations - number of ways to pay

Question: 22 people go to a movie theater. 11 of them are carrying a 50 dollar bill while the other 11 are carrying a 100 dollar bill. The ticket for the movie theater costs $50. The cashier ...
-4
votes
1answer
55 views

Seat 5 men and 4 women in a row such that the women occupy even places [closed]

It is required to seat 5 men and 4 women in a row such that women occupy even places. How many such arrangements are possible?
6
votes
3answers
96 views

Textbooks on permutation groups?

I need good texts on group theory that cover the theory of permutation groups. I think there is a book called Wielandt. Is it good? are there newer alternatives? Can I find books that are not ...
0
votes
2answers
50 views

Confusion in combinatorics

Question (1) The number of different ways in which $10$ telegrams can be distributed to 2 message boys is ____? The answer as per the book is $2^{10}$. But, I think answer should be $10^{2}$. If ...
-1
votes
2answers
35 views

How many ways to divide 11 people into 3 groups of 3 and one group of 2? [closed]

How many ways are there to divide 11 people into 3 groups of 3 and one group of 2? The right answer is 15400 but I can't get it
2
votes
1answer
29 views

Permutations and combinations - choosing an integer

Question: In how many ways can we choose 2 distinct integers from 1 to 100 such that the difference between them is at most 10? Approach: I tried to fix a certain number, and then find the ...
0
votes
1answer
32 views

What's the sign of $\det\phi'=\pm 1$ where $\phi:\mathbb{R}^n\to\mathbb{R}^n$ is a permutation of coordinates?

Let $S_n$ denote the symmetric group and $$\phi:\mathbb{R}^n\to\mathbb{R}^n\;,\;\;\;x\mapsto\left(\begin{array}{c}x_{\pi(1)}\\\vdots\\x_{\pi(n)}\end{array}\right)$$ for some $\pi\in S_n$. Obviously, ...
0
votes
1answer
373 views

Count permutations with LCM

Given $N,M$ and $D$ we need to count how many permutations of $N$ integers are there with each $i$'th element $1 \le A[i] \le M$ such that least common multiple (LCM) of all its elements is divisible ...
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0answers
24 views

Permutations and combinations - divisibility of a factorial

Question: Find the largest value of $n$ for which $125!$ is divisible by $6^n$ Approach: I tried to find all the numbers which were a multiple of 6. The number of times such divisors occurred ...
2
votes
2answers
26 views

Permutations and combinations - number of solutions

Question: Find the number of solutions of $x_1+x_2+x_3 = 51$ for $x_1,x_2,x_3$ being odd numbers Not sure how I would even begin this question. It would be simple except for the condition given ...
1
vote
1answer
29 views

Normalizer of a subgroup generated by a cycle.

Let $H$ be the cyclic subgroup of $S_4$ generated by the cycle $(1234)$. What is the order of the normalizer $N$ of $H$ in $S_4$? Give generators for $N$. How do I go about solving a ...
2
votes
2answers
146 views

Permutations and combinations - distributing objects into groups

Question: In how many ways can $2n$ people be divided into $n$ pairs? Approach: Well as there are $2n$ people, it is obvious that we need to chose each and everyone of them. Using simple ...
1
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0answers
48 views

25 books - permutation question

Imagine 25 books, 5 groups of books (e.g. maths, biology, history, geography, philosophy...) and all groups has 5 seperate colors of books (black, blue, yellow, red, green). So there are 5 blue books ...
3
votes
1answer
36 views

Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...