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

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permutations in races

I am aiming to find the number of different permutations of 3 in 10 races of runners and also the number of permutations which are likely to contain the 3 correct winners. There are 10 races numbered ...
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2answers
28 views

Why must $A_n$ be generated by the 3-cycles

For my course in Group Theory, I have seen various proofs that show why the alternating group $A_n$, which consists of the elements of $S_n$ that can be expressed as an even number of transpositions ...
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1answer
55 views

Permutations with strictly increasing cycles

Find the number of elements of the symmetric group $S_N$ that when decomposed into cycles, have strictly increasing numbers within each cycle. For example for $S_3$, there are 5, only (1 3 2) is ...
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1answer
158 views

Permutation of MATHSBOMBE :

Of all the possible word formed by the permutation ( including non-sensical word ). If you were to arrange all the words in alphabetical order, what would be the position of the word 'MATHSBOMBE ' ...
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2answers
35 views

Finding all the combinations of ways to slice the presidential vote

I'm a journalist who's trying to scope out how many ways a theoretical presidential vote could be split among n candidates. The vote must add up to 100. Each candidate can score between 0-100. So ...
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4answers
279 views

Find the total number of ways in which A can win this series of games

Two players A and B plays a series of $2n$ games. Each game can result in either a loss or win for A. Find the total number of ways in which A can win this series of games. (All games are to be ...
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1answer
43 views

If $G_{\alpha} \cong S_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$. Then $G$ has transitive normal subgroup of index $2$.

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
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24 views

Is $C_{S_n}(g) = H \times S_m$?

Let $g$ be an element of the symmetric group $S_n$. If $c$ commutes with $g$, then $c$ permutes the set of $g$-fixed numbers in $\{1,\ldots,n\}$. Write $C_{S_n}(g)$ for the centralizer of $g$ and ...
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1answer
14 views

Mistake done in this P&C of seating people.

Q. Let 12 seats be occupied by 4 people such that there must be at least two empty seats between any two person. Number of possible arrangement is My approach: Arrange four people now put two-two ...
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1answer
28 views

Significance of the notion of equivalent actions vs. permutation isomorphic action

Let $G$ be a group acting on $\Delta$, and $H$ be a group acting on $\Gamma$. If there exists an isomorphism $\varphi : G \to H$ and a bijection $\psi : \Delta \to \Gamma$ such that $$ \psi( \alpha^g ...
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29 views

Two actions that should be non-equivalent on $A_4$, but they seem to be equivalent?

I was trying to find some actions that are permutation isomorphic, but not equivalent. See my recent post here for the definitions. One natural candidate seems $A_4$. As the subgroups $U_1 = \langle ...
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1answer
6 views

If $\alpha$ is an n-cycle, then $\alpha^k$ is a product of $(n,k)$ disjoint cycles, each of length $n/(n,k)$

If $\alpha$ is an n-cycle, then $\alpha^k$ is a product of $(n,k)$ disjoint cycles, each of length $n/(n,k)$ What I tried is let $a_i$ be an element of the cycle and $l$ be length of each disjoint ...
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1answer
17 views

The product of two transpositions is a commutator

Let $n \ge 4$ and $u,v \in S_n$. Prove that $uv$ is a commutator. In other words, prove that there are $\alpha, \beta \in S_n$ such that $uv = \alpha \beta \alpha^{-1} \beta^{-1}$. This is ...
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1answer
23 views

Example in which a normal subgroup acts non-equivalent on its orbits

Let $G$ be a group acting on $\Delta$, and $H$ be a group acting on $\Gamma$. If there exists an isomorphism $\varphi : G \to H$ and a bijection $\psi : \Delta \to \Gamma$ such that $$ \psi( \alpha^g ...
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3answers
29 views

Number of 5-digit numbers

Question: Construct 5-digit numbers from the digits $0, 1, 2, 3, 4$. Repetition and $0$ at the beginning isn't allowed. a) How many 5-digit numbers can be formed? b) How many of these ...
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32 views

Name for a group action which is almost regular

Let $G$ be a group acting on a set $X$. We say that the action of $G$ on $X$ is regular if for any two elements $x,y \in X$ there exists exactly one element $g \in G$ such that $gx = y$. Is there a ...
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If $G_{\alpha} \cong A_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$ and $V \le G_{\alpha}$ is the four-group in $A_4$, then $C_G(V) = V$

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
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1answer
27 views

Show that every $\sigma \in S_n$ is of the form $\sigma = \prod_i (1 \; \; x_i)$

Let $n \in \Bbb N$ and let $S_n$ denote the group of permutations of $\{1,2,...,n\}$. Prove that for all $\sigma \in S_n$, we have: $$\sigma = \prod_{i=1}^m (1 \ \ x_i), \text{ for some ...
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3answers
32 views

Permutations of variable size using the letters of a word

I have a word, for example SANTACLAUSE and want to calculate all permutations of variable length, but each letter can only be used as often as it it used in SANTACLAUSE. For length = 11 its ...
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25 views

Discussion of $Z(A_4) = \{e\}$

I tried to answer the following question: Why does the fact that the orders of the elements of $A_4$ are $1,2$ and $3$ imply that $|Z(A_4)|=1$? My answer: Two cycles commute if and only if ...
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0answers
21 views

What is the probability of shuffling a deck of cards, & coming up with an order of cards that doesn't have any 2 cards of the same value back to back? [duplicate]

This is a question that I have been struggling with for quite some time. I work with kids at a math tutoring center, and one of the things we do is work on the multiplication facts. We use a ...
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2answers
32 views

Proof for conjugate cycles

Let $\alpha =(a_1,a_2,...,a_s)$ be a cycle and $\pi$ a permutation in $S_n$. Then $\pi\alpha\pi^{-1}$ is the cycle $(\pi(a_1),...,\pi(a_s))$. I'm having trouble understanding the proof of this: Take ...
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1answer
22 views

Multiplication principle and permutation

Definition of Multiplication principle in Principles and Techniques in combinatorics by Chuan-Chong, Khee-Meng is given as: Let $$\prod_{i=1}^rA_i=A_1\times\dots\times A_r=\{(a_1,\dots,a_r) | ...
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1answer
29 views

Solving the equation relating to permutations $^{2n}P_3 =^{6}P_n$

How would you solve this equation? So far i can only do LHS= $^{2n}P_3=\frac{(2n)!}{(2n-3)!}=(2n)(2n-1)(2n-2)$ as I cancel out the $(2n-3)!$ For the RHS I arrived at ...
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3answers
132 views

Permutations and arrangements of toy animals

The question is: A baby has nine different toy animals. Five of them are red and four of them are blue. She arranges them in a line so that the colours are arranged symmetrically. How many different ...
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4answers
49 views

Permutations- the number of six digit integers that are even

Determine the number of six digit integers in which no digit may be repeated and the integers are even. I understand how to do this when we are repeating digits $9*10^4*5$ When repetition is not ...
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3answers
80 views

Permutations- How many three letter words can you produce from the letters BALLSY?

I assumed the way to tackle this problem would be $\frac{n!}{(n-r!)(2!)}$ to account for the two l's which would result in $\frac{6!}{3!2!}=60$ However, when I enumerate the different ...
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1answer
59 views

If $N_G(U) = TU$ for $T = \langle t \rangle$ with involution $t$, and $N \cap U \ne 1$, then $G = TUN$ and $UN$ is Frobenius group

Let $U \le G$ be a subgroup of the finite group $G$ of odd order such that $|N_G(U) : U| = 2$ and different conjugates of $U$ intersect trivially, i.e. $U^g \cap U = 1$ for $g \notin N_G(U)$. Suppose ...
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3answers
53 views

Number of ways so that exactly one permutation of the word TIDE occurs

I want to calculate the number of $8$ letter words that can be formed using the letters of the word $TIDE$. However, in any word only one permutation of the word $TIDE$ should be present. That means ...
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3answers
39 views

In how many ways $A$ speaks before $B$ and $B$ speaks before $C$

$10$ persons has to give a speech among which three are $A$, $B$ and $C$. In how many ways can they give speech so that $A$ speaks before $B$ and $B$ speaks before $C$. I have taken the fixed speech ...
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If $G$ has cyclic Sylow $2$-subgroups, then the core $O(G)$ acts transitive.

Let $G$ be a finite, transitive permutation group on $\Omega$, and assume the point stabilizers have even order. Denote by $O(G)$ the largest normal subgroup of $G$ whose order is odd (see here for ...
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If $G$ acts such that $\mbox{fix}(g) \in \{0,3\}$ for $g \ne 1$, and stabilizers are t.i. subgroups, then the Sylow $3$-subgroups have maximal class

Let $G$ be a transitive permutation group such that every nontrivial element fixing some point fixes exactly $3$ points. Also assume that for $g \notin N_G(G_{\alpha})$ we have $$ G_{\alpha} \cap ...
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+50

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
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Permutations and combinations textbook recommendations

I have had real difficulty with permutation/combination questions in probability and statistics texts. What I have real difficulty with is transforming word problems into mathematical form to solve. ...
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2answers
25 views

Combinatorics question- 8 football games- how many end with 4 wins, 3 losses and a tie

A college plays 8 football games during a season. In how many ways can the team end the season with 4 wins, 3 losses and a tie? I started this question by trying to count the total number of possible ...
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Qualities of Even Permutation Groups

I need some help figuring out some qualities of even permutation groups. Consider $E_n$ to be a subset of the bijection set $S_n$ (bijections over $[n]$) that consists of all even permutations. I want ...
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1answer
19 views

Proving Property of Certain Permutation Groups

I'm trying to show that there is no $g$ such that $g^{-1}(1,2,3)g = (1,3)(5,7,8).$ I am having some general issues figuring out where to move with this problem, since it seems difficult to figure out ...
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1answer
54 views

How many move paths on a 2-d grid? [duplicate]

On a 2-d grid how many different move paths can be made that begin at $(0,0)$ and end at $(x,y)$ with $x\gt 0$,$y\gt 0$. Restriction : left,down and diagonally moves aren't allowed .
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1answer
24 views

Finding the number of objects in permutation [closed]

What is n in this permutation, P(n, 3) = 60? Please help me solve this.
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1answer
41 views

Permutation problem: create words from letters

I'm stuck on this problem: Consider the five letters A, B, C, D, and E. How many words with four letters can you create if each letter can be used at most two times? (One letter can i.e. be used ...
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42 views

Fast way to multiply permutation groups?

I'm having some trouble with permutation group multiplication. When multiplying permutation groups, I always follow the method described in this post: multiplication on permutation group written in ...
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2answers
80 views

Domino tiling extended in N dimensions.

The standard domino tiling problem, is the number of ways to tile a board of size 2xn by dominos of size 2x1. The answer directly follows a recursion, the same as the Fibonacci series. If I extend ...
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75 views

Permutations of given set. [duplicate]

Given a finite set of elements $\{a,b,c,d,e,..\}$ what will be the total number of permutations if sets of two elements each cannot occur side by side (or sequentially in order or in reverse order). ...
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33 views

Number of Sitting Permutations? [duplicate]

There are N persons. You are given M numbers of pairs from these N people. Find total no of permutations to arrange these people in a line in such a way that none of the metioned pair sit together. ...
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29 views

Permutations to divide a solid

I have a 3-dimensional cuboid, with dimensions 2x2x1. I wish to divide this into smaller EQUAL sized cuboids of size 2x1x1. Then I want to extend this case for larger cuboids, assuming that equal ...
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1answer
29 views

Left hand glove and Right hand glove

There are n pairs of gloves and n men. In how many ways can each of the n men have a left hand of one pair and a right hand of another pair of gloves. I thought its a simple question of derangements ...
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3answers
87 views

Can there be a single game of Chess which includes all the possible situations that may arise during Chess?

First of all, I am sorry as there were some completely inappropriate posts posted by my account earlier. This happened because I forgot to log my account out from the computer in Net Cafe I last ...
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1answer
199 views

Arrange $n$ people so that some people are never together. [duplicate]

We have $n$ number of people and some pairs given. These pairs of people are never to be together. How to calculate the number of arrangements possible? e.g we have n=4 and pairs =3 and pairs are ...
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0answers
43 views

How to calculate $a_5,a_4=$

Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,3...n\}$ such that $\sigma $ is a product of exactly two disjoint cycles .Then $a_5,a_4=?$ Calculating $a_4$ :Possible cases which ...
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1answer
59 views

How many ways to line up n objects with distinct heights

Over winter break, I have been working on a few programming questions and I came across this one, which has me a bit stumped: As you ponder sneaky strategies for assisting with the great rabbit ...