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

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Permutations and generators and matrix representations.

I am considering all possible permutations of 3 elements and I want to construct these using the generating elements (switch 1 and 3) and (switch 1 and 2). We can represent these using the matrices: ...
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2answers
30 views

Find The Number Of Permutation

I was solving one probability question, and to solve the problem I could relate it to binary and want solution. The question was that in a fair toss repeated 10 times, what's the probability of 10th ...
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0answers
11 views

Possible Combinations of 3 Digits into 10 [on hold]

Please list all possible combinations of 3 digits: 1,2,x into 10 places
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0answers
20 views

Some subset is not a block in group action iff a separation property holds, questions on proof and special cases

Let $G$ be a group acting transitiviely on a set $\Omega$. A nonempty subset $\Delta$ of $\Omega$ is called a block for $G$ if for each $x \in G$ either $\Delta^x = \Delta$ or $\Delta^x \cap \Delta = ...
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1answer
14 views

The number of quadrilaterals formed from collinear and non-collinear points.

There are $25$ points on a plane of which $7$ are collinear . How many quadrilaterals can be formed from these points ? I did this $^{25}C_{4}-^{7}C_{4}=12615$ quadrilaterals. But the book is ...
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2answers
37 views

Permutation of natural numbers 1 [on hold]

This question belongs to permutation chapter here we have to find the value of $n$. where we are given with the following equation. ...
0
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1answer
18 views

Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.

Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition. proof: Let $\sigma \in S_n$. Then $\sigma = ...
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1answer
13 views

$f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$?

Let $f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then is it true that $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$ ? Since both $f,g$ are non-trivial , I ...
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0answers
15 views

Number of sets containing m decomposable permutations of n objects.

Let $P_{m,n} = \{ \sigma_i \in S_n \}$ be a set containing $m$ arbitrary permutations of $n$ objects. Let $Q_{m,n} = \{\sigma_{ij} = \sigma_i^{-1}\sigma_j \mid \sigma_i, \sigma_j \in P_{m,n} \}$ be ...
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1answer
33 views

Find isomorphism between $S_3$ and $GL_2(F_2)$. [duplicate]

Find isomorphism between $S_3$ and $GL_2(F_2)$. proof: Let $A = \begin{pmatrix} a& b\\ c & d \end{pmatrix}$. Where $\det (A) \neq 0$. And recall $S_3$ is the permutation group with ...
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1answer
35 views

Numbers between $200$ and $1200$ that can be formed with the digits $0,1,2,3 $

How many numbers between $200$ and $1200$ can be formed with the digits $0,1,2,3 $ (repetition of digits not allowed ) ? $a.)\ 6\\ b.)\ 8\\ c.)\ 16\\ \color{green}{d.)\ 14}$ I divided it in ...
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1answer
58 views

Permutations on word $MISSISSIPPI$.

In how many ways can the letters of the word $MISSISSIPPI$ be rearranged ? I am confused on whether it is $\dfrac{11!}{4!4!2!}$ or $\dfrac{11!}{4!4!2!}-1$ since it is given rearranged and not ...
2
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1answer
32 views

Permutation of students in a class

In how many ways can 10 BS and 7 MS students be arranged in a line so that no two MS students may sit together? My approach: Total number of ways all 17 students can be arranged in a line is ...
2
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1answer
40 views

Permutations and Combinations - conceptual

Suppose we have 10 objects. I want to create a group with those 10 objects. The group should contain a minimum of 2 objects (it can contain anywhere from 2-10 members). How would I find the total ...
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1answer
23 views

Are cyclic permutations always even?

I just discovered the permutation symbol $\epsilon_{ijk}$ and was wondering if the set of cyclic permutations and even permutations are the same.
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2answers
41 views

Capable groups of order $32$ with GAP

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. How can I find all capable groups $G$ of order 32 with $|Cent(G)|=10$, where $Cent(G)$ is the set of all ...
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3answers
37 views

15 indistinguishable fishes be placed into 5 different ponds

In how many ways can 15 indistinguishable fishes be placed into 5 different ponds, so that each pond contains atleast one fish? I am struck on this problem.Can someone help me out please.
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1answer
24 views

Distance between two ordered sets

Is there a way to measure the "distance" between two ordered sets? Say i got two sequences of letters: $$ S_1 \{A, B, C, D, E, F\} $$ $$ S_2 \{B, C, D, A, F, E\} $$ How could I find an "amount of ...
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1answer
18 views

Cardinality of the set $D$

Let , $D$ be the set of tuples $(w_1,w_2,\cdots,w_{10})$ , where $w_i \in \{1,2,3\},1\le i\le10$ and $w_i+w_{i+1}$ is an even number for each $i$ with $1\le i\le 9$. Then find the cardinality of ...
2
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1answer
29 views

How to solve this kind of combinatorics problem?

I have a question about combinatorics. Here is the question: A waiting area outside the doctor's office contains a row of 7 chairs. In how many different ways can a man, a woman and a boy occupy 3 ...
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4answers
27 views

No: of ways to distribute cards .

In how many ways can a person send invitation cards to $6$ of his friends if he has $4$ servants to distribute the cards ? $a.)\ 6^{4}\\ \color{green}{b.)\ 4^{6}}\\ c.)\ 24\\ d.)\ 120$ As the ...
2
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0answers
47 views

Combinatorics: Permutation Problem, how to know if a solution is correct or wrong

Question: Find the number of ways of arranging 8 Men and 2 Women in a row such that 2 Women are never together. For the above question, I thought of 2 ways to proceed 1> Arrange 8 men in 8! ...
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4answers
77 views

Combinatorial proof $\sum_{i=1}^n i/(i + 1)! = 1 - 1/(n+1)!\quad\forall n\in\mathbb N$

I am trying to come up with a combinatorial (or at least partly combinatorial) proof of the equation $$\sum_{i=1}^n \frac i{(i + 1)!} = 1 - \frac 1{(n+1)!}\quad\forall n\in\mathbb N$$ I am thinking ...
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1answer
40 views

Proof of stars and bars formula

I am trying to prove a formula (for ways of distributing n identical balls among r persons when each person may get any number of balls) C(n+r-1, r-1). But I am not able to prove it. I may be doing ...
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1answer
26 views
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3answers
24 views

how many ways a captain be chosen? [on hold]

from a group of $40$ players a cricket team of $11$ players is choosen. Then one of the $11$ is choosen as the captain of the team. How many ways this can be done ?
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0answers
74 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
0
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1answer
33 views

Finding all homomorphisms [duplicate]

How can I solve this question Find all group homomorphisms from $\mathbb{Z_{5}}$ into $\mathbb{Z_{12}^{x}}$?
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1answer
26 views

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart [closed]

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart??
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1answer
36 views

Showing a subset of $S_n$ is a subgroup

Let $P$ be the set of all the elements of $S_n$ which can be written as $\sigma\mu\sigma^{-1}\mu^{-1}$ for $\sigma, \mu \in S_n$. Show this is a subgroup. This doesnt seem to be as simple as ...
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0answers
30 views

How can I formed as below permutation problem

Hi I am writing a program and i encouraged the below permutation problem and need your help. There are 4 boxes: 3 of them have 2 balls The one box has 1 balls. The question is what is the ...
2
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1answer
36 views

count permutations that do not contain repeated combinations

I am trying to count the number of permutations that do not contain order specific groupings that have occurred in permutations that have already been counted. Example: For the set {A B C D E}; if ...
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0answers
15 views

Number of injective maps from one finite set to another [duplicate]

$m\le n$ be natural numbers. What is the number of injective maps from a set of cardinality $m$ to a set of cardinality $n$ $?$ I think it is the number of ways $m$ ...
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1answer
17 views

Random Permutation Polynomial With Fixed Inputs

Assume we pick uniformly random a permutation polynomial, $T$, of degree one. we define all polynomials over $\mathbb{Z}_P$. We have fixed inputs $x_i$ (e.g. $x_i \in [1,100]$) My Question: Is ...
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0answers
28 views

Calculating Combinations / Permutations [closed]

How do I calculate the number of outcomes as a whole of a series of individual tests with there own outcomes? For example, the best description I could think of would be: There are 10 tests and each ...
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1answer
16 views

Proving that an adjacency transposition is the product of odd number of adjacencies.

A transposition in $S_n$ of the form $(i \ i + 1)$ is called an adjacency. I am trying to prove that, Given $i ∈ \{1, . . . , n − 1\}$, if $i < j$, the transposition $(i \ j)$ is a product of an ...
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0answers
40 views

I need the paper “ J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365” [closed]

I need this paper " J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365", But i not found. Who can help me to find this paper ?
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0answers
48 views

identity permutations [closed]

I need some help with the following question : For the permuation $ π $ on n elements we define the term : $ π^k=i $ if the composition of π on it self k times is the identity permutation . (where ...
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0answers
24 views

how many transpositions exist in this permutation? [closed]

For permutation \begin{pmatrix}1&...&l&l+1&...& l+k\\k+1 & ...&k+l&1&...&k\end{pmatrix}. The answer is kl. Thanks
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1answer
22 views

Reason of dividing to n! ( repetition ) on Permutations with Repetitions

I'm trying to figure out the reason of diving the number of permutations by the number of repetitions (in factorial). Shouldn't it be without the factorial? I don't get why are there is a factorial in ...
2
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1answer
33 views

In how many ways can five-digit numbers be formed by using digits $0,2,4,6,8$ such that the numbers are divisible by $8$?

In how many ways can five-digit numbers be formed by using digits $0,2,4,6,8$ such that the numbers are divisible by $8$? Assume the case in which repetition is not allowed Our Approach: Case1: ...
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0answers
40 views

How to find the number of solution of $x^n=1$ in the group $S_n$?

Suppose that $S_n$, the symmetric group of order $n!$ is given and for given $m\in \mathbb N$ fixed, we are to find the number of solutions to $\theta^m=e, \theta\in S_n$. Can someone tell me or give ...
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0answers
66 views

the identity permutation [closed]

for the permuation $ \pi $ on n elements we define the term : $ \pi^k=i $ if the composition of $ \pi $ on it self k times is the identity permutation . A. let $ a_n $ be the number of permutation of ...
2
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2answers
44 views

How many 4 digit numbers can be formed using numbers 2,3,4,5,6,7 such that the number is only once divisible by 25?

Q How many 4 digit numbers can be formed using numbers 2,3,4,5,6,7 such that the number is only once divisible by 25? My approach: Case1: Unit digit is 5.Ten's digit will be 2 or 7.Taking here 2 ...
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2answers
21 views

If abc is a three digit number such that not one number is similar to other than how many possible values of ( a + 4b + c ) will be divisible by 40?

If abc is a three digit number such that not one number is similar to other than 1 How many possible values of ( a + 4b + c ) will be divisible by 40? 2 How many possible values will be ...
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0answers
25 views

Permutations of given length

Given the counts of each of three letters a, b, and c. I want to find all permutations of a given length where each letter can occur at most times as the given count. I am only interested in the ...
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1answer
40 views

How many 3 digit no.s can be formed so that the sum of two digits will be equal to the third digit?

Qn How many 3 digit no.s can be formed so that the sum of two digits will be equal to the third digit? Q I am confused in this question whether to take first 2 digits sum or any digit sum such ...
4
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3answers
77 views

How many 3 digit numbers can be formed using digits 1,2,3,4 and 5 such that the number is divisible by 6

How many $3$ digit numbers can be formed using digits $1,2,3,4$ and $5$ without repetition such that the number is divisible by $6$ First Approach: A number is divisible by $6$ if it is ...
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2answers
27 views

How many combinations are possible, if fare is 50 paisa?

Sherlock Holmes and Dr Watson travel from X to Y via metro. They have enough coins of 1,5,10,25 paisa. Sherlock Holmes agrees to pay for Dr Watson only if he tells all possible combinations ...
0
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1answer
64 views

Permutations excluding repeated characters

I'm working on a Free Code Camp problem - http://www.freecodecamp.com/challenges/bonfire-no-repeats-please The problem description is as follows - Return the number of total permutations of the ...