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

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2
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1answer
26 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
4
votes
1answer
15 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
3
votes
1answer
32 views

How many permutations of the letters DANIEL do not begin with D or do not end with L?

How many permutations of the letters DANIEL do not begin with D or do not end with L? The correct answer is 696. This answer does not make sense as there are 120 (5!) ways the letters can be ...
0
votes
1answer
30 views

In how many ways can the committee be selected if the girls must include either Roberta or Priya but not both?

A committee of three boys and three girls is to be selected from a class of $14$ boys and $17$ girls. In how many ways can the committee be selected if the girls must include either Roberta or Priya ...
0
votes
2answers
23 views

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 ...
1
vote
2answers
29 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 ...
1
vote
2answers
60 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 ...
2
votes
2answers
109 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 ...
3
votes
4answers
86 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 ...
0
votes
2answers
21 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
42 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 [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 ...
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
36 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
103 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 ...