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

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Permutations and Combinations Tricky Question

In a photo there are three families (six Greens, four Browns, and seven Grays) arranged in a row. The Browns have had an argument so no Brown will stand next to another Brown. How many different ...
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2answers
46 views

Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
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2answers
77 views

How many ways are there of splitting twelve people into two groups of the same size?

Twelve people need to be split up into teams for a quiz. How many ways are there of splitting them into two groups of the same size? I did $12 C 6$, which gives $924$, however the answer is ...
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2answers
125 views

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet?

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet? I tried $5 C 1 + 5 C 2 + 5 C 3 + 5 C 4 = 30$, however, the answer is $20$. Any ...
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5answers
102 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
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22 views

Combination or Permutations - identification

Car $A$ can take $5$ passengers, car $B$ can take $6$ passengers, car C can take $2$ passengers. Find the number of ways that $11$ passengers and a couple to travel in the $3$ cars. $${13 \choose{5}} ...
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1answer
25 views

Inverse Permutations from $S_7$

Would someone mind giving an explanation of how to find the inverse permutation of: $(1 2 3 5 7)^{-1}$ in $S_7$? I am not quite understanding how to do this.
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0answers
29 views

Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
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0answers
43 views

Number of permutations on nearest neighbors

Consider a finite connected set $A \subset \mathbb{Z}^d$ and let $S_A$ be the set of permutations on nearest neighbors. Namely, the elements of $S_A$ are bijections $\pi : \, A \rightarrow A$ such ...
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1answer
23 views

Best algorithm for finding permutations with constraint of average total value.

Let's assume I have a random number generator from 0-100 included (only integers) and I generate 5 numbers with it. I want to know the probability of hitting 80, 80, 80, 80, 80 with the constraint ...
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1answer
28 views

Optimize order of a list based on time to complete, probability of success

I'm a programmer participating in a coding challenge, but I'm not up on my advanced math. I'm currently working on a solution to a problem, and have a semi-functional program, but I'm still missing a ...
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1answer
44 views

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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0answers
25 views

Permutation calculator

I am studying the Mathieu group $M_{12}$ on the twelve letters $\infty,7,6,8,X,2,0,3,4,1,9,5$ (in this specific order) in the form that it is generated by the permutations $(0123456789X)$, ...
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1answer
28 views

Combination during time period

There is a workforce who can handle $3$ products and the $3$ products have different execution times: $1h$, $2h$, $4h$. How do I calculate all possible combinations this workforce may create between ...
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2answers
36 views

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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1answer
30 views

Diagonal elements of subset of Hadamard matrices

I'm looking at Sylvester's construction of Hadamard matrices, where $H_{2^n} = \left[\begin{array}{c c} H_{2^{n-1}} & H_{2^{n-1}} \\ H_{2^{n-1}} & -H_{2^{n-1}} \end{array}\right]$, where ...
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0answers
32 views

How do I calculate all possible combinations for a player creator in a game?

I'm currently working on a character creator for a game, but I don't know how to calculate all possible character combinations the player can create. In the creator, the player is required to choose ...
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2answers
47 views

Arrangement of all the letters of a word [closed]

In how many ways can all the letters of the word ‘PERFORMED’ be placed in the $3 \times 3$ grid of squares, such that each square contains exactly one letter and there is at least one vowel in each ...
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1answer
78 views

The Rubik Square permutation groups

This post was inspired by this webpage of mathematical challenge due to Mickaël Launay (French). Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as for the following ...
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2answers
44 views

Prove image of symmetric group into additive group of real numbers is zero

Suppose $ f : S_{n} \rightarrow (\mathbb{R} , + , 0 , - )$ is a group homomorphism. Prove $ f(S_{n}) = {0} $, i.e., $f(\sigma) = 0$ for every $ \sigma \in S_{n}$with $n \geq 1 $ I cannot seem to find ...
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1answer
39 views

Balls and Boxes Generalization

Recently, I saw a problem here on MSE: $$$$"Put 9 pigs in 4 pens such that there are an odd number of pigs in each pen." Individual cases or solutions to the problem are quite easy. But how would we ...
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1answer
36 views

What does it mean to find elements in $S_9$ that are “not cycles”?

I came across this wording in the following question. Some clarification on what this means and how to approach this problem would be helpful. Thanks!
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2answers
43 views

Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
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1answer
118 views

How many ways to add to 32?

I have been presented with a rather complex combination problem. Using only the numbers 2, 4, 6 and 8, how many possible ways can you add up to 32 if the number 4 may only be used no more than once ...
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1answer
20 views

Number of ways of selecting all k-indexed identical items before all k+1 indexed identical items for all k from 1 to n

Suppose we have n indices and we have a specific number of items allotted to this index. Say for 2 balls of colors Blue(B)[1], 4 of color Green(G)[2] and 2 of color Red[3] (I could've just assigned ...
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2answers
45 views

Should I be using combinations or permutations?

I have a set of $26,000$ values. Each value has the option of being $1$ or $0$. How do I calculate the number of potential combinations of $1$'s and $0$'s that exist for $26,000$ values?
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0answers
23 views

Invertible matrices, permutations and leading principal minors

Given an invertible $\{-1,0,1\}$-matrix $A$ (its determinant is $\pm 1$), are there two permutation matrices $P$ and $Q$ such that all the leading principal minors (determinants of the top-left ...
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0answers
27 views

How many symmetries does the Cauchy-Schwarz inequality have?

The symmetries of the Cauchy-Schwarz inequality $(x_1^2 + \cdots + x_n^2)(y_1^2 + \cdots + y_n^2) \ge (x_1y_1 + \cdots + x_ny_n)$, as a subgroup of the symmetric group on the $2n$ letters $x_1,\ldots, ...
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2answers
59 views

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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2answers
38 views

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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1answer
25 views

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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1answer
33 views

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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1answer
28 views

transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
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30 views

In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least one ball?

In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least 1 ball? I proceeded like this- In the first row choosing one cell out of ...
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2answers
31 views

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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1answer
23 views

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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1answer
21 views

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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1answer
27 views

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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2answers
22 views

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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1answer
8 views

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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0answers
16 views

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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1answer
13 views

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 ...
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32 views

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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61 views

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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4answers
57 views

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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2answers
53 views

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. : ...
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3answers
38 views

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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1answer
30 views

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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2answers
52 views

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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2answers
13 views

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 ...