# Tagged Questions

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

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### Permutations for a set of rules

The question is from - http://www.iarcs.org.in/inoi/2015/zio2015/zio2015-question-paper.pdf - Q.2 I tried solving it but I have no clue how to go about doing it. The question says that a railway ...
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### Does transitive imply it's the entire symmetric group

Let $G$ denote a finite group and recall that $G$ acts transitively (on itself) if and only if for all $x,y \in G$ there is a $g \in G$ such that $gx = y$. I am wondering if transitive may imply that ...
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### Applying Inclusion-Exclusion principle

How to apply principle of inclusion-exclusion to this problem? Eight people enter an elevator at the first floor. The elevator discharges passengers on each successive floor until it empties on ...
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### How many combinations can you get from a three times three matrix

I have a 3*3 matrix like this (figure 1): * * * * * * * * * Slots can be filled similar to next examples (figure 2): ...
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### Permutation of numbers that there are all modulo M .

Let's say I have $M-1$ integers, all of them different from each other, and all of them smaller than integer M: $$1,2,3...M-1$$ I multiply each of them by another integer S, and write the result ...
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### How to use class equation for determining the center of $S_4$

How to use class equation for determining the center of $S_4$ $$|G|=|Z(G)|+\sum_x [G:C_G(x)]$$ So I guess I need to find $$|G|-\sum_x [G:C_G(x)]=|Z(G)|$$ Well $|S_4|=4!=24$ and $C_G(x)$ is the set ...
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### Product of disjoint cycles and product of transpositions

$\alpha=(3412)(245)\in S_5$ and I have to 1) write this as a product of disjoint cycles, 2) write this as a product of transpositions. 1) I can do thing by following where the elements go in the two ...
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### can i do this transformation with any finite group?

I have a finite alphabet $\{e_1, e_2, \cdot \cdot\cdot, e_N, a_1, a_2, \cdot\cdot\cdot, a_n \}$ where we pair $e_i$ and $a_i$ as opposites'' - like opposite vertexes on a regular $2N$ sided polygon ...
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### Understanding representation of permutation matrix as vector

I hope this question is relevant here: I'm using some external software that does an LU decomposition of a square $(n\times n)$ matrix; the result is returned as three matrices L, U and P where P is ...
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### permutations and probabilty

In a certain country, the number plate on a car consists of any 3 letters of the alphabet (the first letter is always a "K" or a "G"), followed by any 3 digits (0 to 9) and a alphabet. For a car ...
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### Permutation with constrained repetititons

The question is as follows: How many ways can 12 identical white and 12 identical black pawns be placed on the black squares of an 8 x 8 chessboard My answer was $\frac{32!}{12!*12!}$ But the ...
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### Rewrite permuatation as disjoint cycles

Rewrite $(3412)(245) \in S_4$ as a product of distinct cycles. I've only ever been given permutations as distinct cycles, transpositions or the matrix notation so I have no idea where to start.
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### Rook Polynomials with Symmetrical Overlap (Count Permutations Restricted by Distance)

Consider the cardinality $P(n,d)$ of permutations where elements can move up to distance $d$; for example, the permutation $\binom{012}{102}$ with $d = 1$ would be valid but $\binom{012}{201}$ would ...
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### How to find all possible groups of four different values(integers)

I have four values :50,100,500,1000. I want to know many groups could be made with this combinations values. 50,100,500,1000 here it would be count as 1+1+1+1 50,50,50,100 count= 3+1 50,50,...
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### Normal subgroup in S4 [duplicate]

Let H be a subgroup of S4 where $H = \{e, B , C ,D \}$ $B(1)=2,B(2)=1,B(3)=4,B(4)=3$ $C(1)=3,C(2)=4,C(3)=1,C(4)=2$ $D(1)=4,D(2)=3,D(3)=2,D(4)=1$ Prove that H is a normal subgroup. I've tried ...
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### How do I find the probability of some elements being together inside a randomly arranged set?

If I have a total of $n$ balls made of $k$ red balls and $(n-k)$ green balls and I arrange them all randomly in a line, how can I calculate the probability $x$ of a group of $y$ red balls being ...
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### Sum of Digit Permutations

The question simply states "Let a secret three digit number be $cba$. If the sum of $cab + bac + bca + abc + acb = 2536$, what is $cba$?" I have no idea how to approach this problem. Any hints or help ...
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### How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed. [closed]

Q. How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed. A. $21$ Answer is correct (please provide a thorough explanation). Unit digit nos. : $3$...
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### We are given a class consisting of 4 boys and 4 girls.

We are given a class consisting of 4 boys and 4 girls. a committee that consists of a President, a Vice-President and a secretary is to be chosen among the 8 students of the class. Let a denote the ...
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### The number of possible combination of column values with possibly common elements

I would like to calculate possible combinations for a given set of data: There is an x amount of columns (let's say 3) each column contains y amount of words (lets say 2), now I would like to ...
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### Permutation of 6-digit numbers without repetition

How many 6-digit numbers without repetition of digits are there such that a ) the digits are all non-zero b ) 1 and 2 do not appear consecutively in either order ? Calculated the answer as below ...
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### How do you find the order of a cyclic group?

What is the order of the cyclic group generated by $(1 2 5)(3 4)$? What is the order of the cyclic group generated by $(1 2 5)(3 5)$? I've looked through my notes and can't find notes on this and can ...
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### Count permutations of $\{1,2,…,7\}$ without 4 consecutive numbers - is there a smart, elegant way to do this?

Here's a problem I've solved: Count permutations of $\{1,2,...,7\}$ without 4 consecutive numbers (e.g. 1,2,3,4). So I did it kinda brute-force way - let $A_i$ be the set of permutations of $[7]$, ...
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### Permuation and Combination: Selecting n numbers such that such that sum is less than m

The value of n can be 0,1,2,3....and so on For example If we have to select 2 numbers such that the sum of all them can be less than 2 Manually the combinations can be (0,0), (0,1), (1,0), (2,0), (...
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### Combinatorics of given alphabet

I'm looking for the formula to determine the number of possible words that can be formed with a fixed set of letters and some repeated letters. For instance take the 8-letter word SEASIDES and find ...
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### How many four-digit odd numbers, all of digits different, can be formed from the digits 0 to 9, if there must be a 5 in the number?

How many four-digit odd numbers, all of digits different, can be formed from the digits 0 to 9, if there must be a 5 in the number? I know that there are 4 different cases where 5 is in the number: ...
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### Cardinality of the set of automorphisms of $(\mathbb{N},+)$

I wonder if the set of bijections $\sigma\,:\mathbb{N}\to\mathbb{N}$ that satisfy $$\sigma(a+b) = \sigma(a)+\sigma(b)\qquad \forall a,b\in\mathbb{N}$$ is countable or uncountable. What if we also ...
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### Even permutations

I am given the symmetric group $S_{9}.$ Let \sigma = \begin{bmatrix} 1 & 2& 3& 4& 5& 6& 7&8 &9 \\ 4& 8& 7& 9& 3& 1& 2& 5 &...
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### Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
### What is the best algorithm for finding a $g \in S_n$ which $a^g=b$ for given $a, b \in S_n$
What is the best algorithm for finding a $g \in S_n$ which $a^g=b$ for given $a, b \in S_n$, where $S_n$ is a symmetric group and $a$ and $b$ have same cycle type? Question 2: Is there any command in ...
Show that for every 2 elements $\alpha$ and $\beta$ in $S_{8}$, the permutation $\alpha ^{-1}\beta ^{2}\alpha$ is an even permutation. How do I show that the above is an even permutation? I know ...