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

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3
votes
4answers
80 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 ...
-1
votes
2answers
17 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}} ...
1
vote
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.
0
votes
0answers
28 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 ...
0
votes
1answer
48 views

Show that $c_n=\frac{n!}{4(n-4)!}$ [on hold]

Let $c_n, n\geq1$ be the number of pair $(\sigma,\tau)$ of permutations $\sigma , \tau \in S_n$ of Type $(1^{n-2},2)$ with the product $\sigma \tau$ of Type $(1^{n-4},2^2)$. Show that ...
0
votes
0answers
32 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 ...
0
votes
1answer
29 views

How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
1
vote
1answer
21 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 ...
1
vote
1answer
20 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 ...
1
vote
1answer
41 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 ...
1
vote
0answers
22 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 $(0123456789X)$, ...
0
votes
1answer
27 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 ...
2
votes
2answers
34 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 ...
2
votes
1answer
29 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 ...
0
votes
0answers
26 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 ...
-3
votes
1answer
20 views

in how many ways can a group of 5 boys and 5 girls be seated in 10 seats with a restriction that 3 girls do not want to sit beside each other [closed]

In how many ways can a group of 5 boys and 5 girls be seated in 10 seats with and the 3 girls do not want to sit beside each other?
-4
votes
1answer
27 views
1
vote
2answers
33 views

Arrangement of all the letters of a word [on hold]

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 ...
2
votes
1answer
67 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 ...
3
votes
2answers
35 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 ...
0
votes
1answer
27 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 ...
-3
votes
0answers
27 views

Suppose a sequence of 8 nucleotides contains 2 each of A, C, G, T. How many such sequences are there? [closed]

Suppose a sequence of 8 nucleotides contains 2 each of A, C, G, T. How many such sequences are there? please post the solution..
-1
votes
2answers
45 views

How many ways can the players enter? [closed]

$5$ players want to enter a stadium through three gates of the stadium, However, each gate of the stadium can only pass two players. How many ways can the players enter the stadium?
1
vote
1answer
35 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!
1
vote
2answers
41 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? ...
-3
votes
1answer
46 views

Lottery Winning probability [closed]

In each lottery game, there are a total $10000$ different combination and only $23$ winning numbers to be chosen. ($0000$ to $9999$) Given the option to randomly choose $500$ different combination ...
7
votes
1answer
102 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 ...
1
vote
1answer
19 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 ...
-1
votes
2answers
43 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?
1
vote
0answers
20 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 ...
2
votes
0answers
25 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, ...
0
votes
2answers
52 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 ...
0
votes
2answers
28 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): ...
-3
votes
0answers
15 views

counting problem for falling numbers [duplicate]

A falling number is an integer whose decimal representation has the property that each digit except the units digit is larger than the one to its right. For example 96521 is a falling number but ...
0
votes
0answers
50 views

on subset of items in ordered list would like to calculate cardinality of set of orderings grouped by kendall tau distance

Let's say I have an ordered list of length $n$ which I will denote $12\ldots n$. There are $n!$ ways to rearrange the items in this list. Take a subset of the items in the list $B\subset ...
0
votes
1answer
20 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 ...
0
votes
1answer
29 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 ...
2
votes
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 ...
1
vote
2answers
25 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 ...
0
votes
2answers
27 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 ...
0
votes
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 ...
1
vote
1answer
18 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 ...
0
votes
1answer
24 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 ...
1
vote
2answers
21 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 ...
1
vote
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.
1
vote
0answers
15 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 ...
0
votes
1answer
11 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 ...
0
votes
2answers
30 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 ...
1
vote
2answers
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 ...
3
votes
4answers
54 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 ...