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

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$G$ is a primitive group

Let permutation group $G$ contains a minimal normal subgroup $\neq 1$ which is transitive and Abelian. Show that $G$ is primitive. My attempts: Because of Proposition 4.4. of Wielandt's book ...
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34 views

Permutation of Groups - looking for the right term

I'm looking for more detailed information about the following problem, but i'm missing a right keyword, or term for this: Let's assume i have 10 people and they are assigned to groups: ...
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34 views

Represent a bijection using a permutation

Let $X = \{1, 2, 3, 4, 5, 6, 7\}.$ For every $n \in X$, write $n^2 - 3n^5 = 7q_n + r_n, 1 \leq r_n \leq 7.$ Define a function $f: X \to X$ by $f(n) = r_n.$ (a) Find an element $\alpha \in S_7$ that ...
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153 views

Arranging books on the shelf.

There are five distinct computer science books, three distinct mathematics books, and two distinct art books. In how many ways can these books be arranged on a shelf if no two of the three mathematics ...
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1answer
25 views

Select K numbers from N numbers fairly

I want to fairly select K numbers out of an array of N number. I know that this problem can be solved using Reservoir Sampling but I want to know if this approach is correct too? ...
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41 views

Permutation (inclusion-exclusion)

2 corrected exams are being returned to each of n students. How many ways can the teacher give those 2 exams back to each student such that everyone receives at least 1 exam that is not his. I know ...
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14 views

How many sequences of length N squared can be formed with N different values where each value is used exactly N times?

For instance, for N=2, the answer is 6 (e.g. aabb, abab abba baab baba bbaa). For N=3, the answer is 1680. I'm looking for the proper formula. Thanks
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17 views

Is there an effective way to convert a product of 2-cycles into a product of n cycles?

I came across this problem that asks me to convert (12)(34) into a product of 5 cycles. After testing for many different combinations i get (12345)(14352)(12345). The way I do it is this: ...
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42 views

Strategy for number of non-negative integers solutions such that $x_1+x_2+\frac{\enspace\enspace\enspace}{}+x_5 = 50$

I'm trying to figure out the number of solutions to the following problems, although I'm not entirely sure what strategy I should use to solve these. Combinations of non-negative integers ...
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2answers
35 views

Permutations and school timetable

If there are 6 periods in each working day of a school. In how many different ways can one arrange 5 subjects such that each subject is allowed at least one period? I tried this way- One of the six ...
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47 views

Permutations in products of disjointed cycles

How do I calculate the following permutation in the symmetric group $S_6$ giving the answers as products of disjoint cycles: $$(2,3,5,6)(1,6,2,4)$$ I have tried following this question but I don't ...
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27 views

Strategies for solving permutations of a word

So I'm trying to prepare for exams, and am having some trouble with permutations, and was wondering what's a good strategy to solve this task is: Given the set of letters $\text{AAABBBBCCDEEFG}$ ...
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2answers
31 views

Number of different possible permutations of a telephone number

A telephone number consists of $10$ digits, all from $0$ to $9$. The first digit is $0$. The remaining digits can be any number ranging from $0$ to $9$. How many possible telephone numbers are there? ...
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35 views

On transitive constituents of a permuation group

Assume that the intransitive permutation group G has degree n and minimal degree n−1. If no transitive constituent of G has degree 1, then they all are faithful and all except one are regular. I ...
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2answers
202 views

Permutations / Combinations - suppose a word is a string of 8 letters of the alphabet with repeated letters allowed

1.) How many words are there? Not sure how to solve this since repeated letters are allowed. $n^r$ is the formula we are told to use for permutations with repeated objects, but $26^8$ seems like too ...
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1answer
24 views

Combinatorics problem

I am trying to solve this question, my solution involves solving a combinatorial problem as follows : Number of arrangements of exactly k distinct elements in n slots such that each one of the ...
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35 views

Concerning cycles and group actions.

Here is the problem that I have. Let $C=\{a=(ijkl)\}$ be the set of all cycles of length 4 in the symmetric group $S_4$. $S_4$ acts on the set $C$ by conjugation. For every cycle $a\in C$ determine ...
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234 views

Combinations of pizza toppings with at least one vegetable and at least one meat.

Here is a question from my quiz: Superior Pizza has seven vegetable ingredients and nine meat ingredients. The number of ways to select five ingredients (no doubling on ingredients) with at ...
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57 views

Combinations - 17 women and 21 men to form a committee of size 7

How many committees are possible if a committee must have $3$ women and $4$ men? $_{38}C_3+_{38}C_4$ or $\frac{38!}{3!35!}+\frac{38!}{4!34!} = 8,435+73,815 = 82,251$ How many committees are possible ...
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1answer
30 views

Permutation - 17 women and 21 men to form a committee of size 7

How many committees are possible? I added the total number of women ($17$) and the total number of men ($21$) to get $38$ total people. I used this as my $n$ or objects. I then subtracted my $r$ ...
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36 views

If $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)?$

For two permutations $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)? \text{ (where }a,b,c,d \text{ are all distinct)}.$ I'm fairly new ...
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1answer
22 views

How many possible permutations are possible if ranking n entities using the 'standard competition ranking' strategy?

I don't know if I'm missing something here, but this doesn't look as straightforward to me as I thought it to be. I basically want to calculate the number of unique rankings that are possible when ...
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2answers
52 views

A sequence $a_i$ such that $|a_1-a_2|,|a_2-a_3|,\ldots$ is also permutation of the positive integers

Let $a_1,a_2,\ldots,$ be a permutation of the positive integers. Is it possible that $|a_1-a_2|,|a_2-a_3|,\ldots$ is also a permutation of the positive integer? My idea is to construct the sequence ...
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1answer
23 views

Arrange numbers to prove question

Prove that 100 0's, 100 1's, 100 2's, 100 3's, 100 4's, 100 5's, 100 6's, 100 7's, 100 8's and 100 9's cannot be used in any form to make a perfect square. I have no idea how to do this question. I ...
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1answer
38 views

Find all possible arrangements of numbers, but keeping the sum constant

How would I find all possible combinations of $n$ natural numbers from 1 to 100, in such a way the sum of the $n$ numbers is always 100. For example if $n = 3$, possible answers would be: $(1, 2, ...
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29 views

Possible choices for coloring boxes with exactly n colors

I have a number of boxes $N$ that I each need to paint with one color chosen from $n$ available colors. Every color must be used at least once and the order in which I color the boxes matters. For ...
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1answer
28 views

combinatorics four digits out of three

With three given digits (1,2,3), how many unique four-digit combinations can be made if all three digits must be present but may be repeated? Example of correct combinations: (1,2,3,3) (1,1,2,3) ...
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1answer
35 views

A combinatorial coefficient linked to exterior product

I am looking at the following sum $$ \sum c_1\wedge \cdots\wedge c_n $$ where the summation ranges over $c_1,\ldots,c_n$ such that each $c_i\in\{a,b\}$ and $a$ appears exactly $j$ times. Thus, using ...
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1answer
47 views

Permutation and combination difference, a flower shop question

I have got a perm/combination question In a certain flower shop, only 3 vases of flowers and 1 wreath can be displayed in the front window at a time. If there are 10 vases of flowers and 4 wreaths ...
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83 views

How many positive integer are there ?? [closed]

My task is to calculate the number of positive integers which are smaller than 10000 and contain “1”. Please give me some hints; thanks in advance.
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98 views

Permutation of 4 letters of the word 'examination'

How many permutations of 4 letters can be made out of the letters of the word 'examination'?
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16 views

permutation -number of numbers of four different digits formed from the digits of the number 12356 such that it is divisible by 4

The number of numbers of four different digits that can be formed from the digits of the number 12356 such that it is divisible by 4??
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67 views

Probability of Permutations/Combinations

How do you set up the formula for the probability of a permutation/combination? Question: If you have a group of candy with $2$ Snickers, $4$ Kit Kats, and $2$ Butterfingers and you take two pieces ...
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1answer
132 views

Permutations and number of permitted combinations three percentages which must add up to 100%

is there a simple way to find the number of combinations of three percentage values with discrete step sizes which add up to 100%? Example: ...
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1answer
26 views

permutations indistinguishable objects and groups

There is a group of 10 objects, 2 red, 3 blue and 5 green. If the 5 green objects should always be placed together, in how many ways we can put them on a line. I did this: As 5 places are occupied by ...
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12 views

combination and permutation with restraint

There is a group of 10 objects, 2 red, 3 blue and 5 green. If the red should be one at the beginning of the line and the other at the end, calculate how many combinations. As the two reds are ...
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1answer
41 views

Combination and permutation of indistinguishable objects

There is a group of 10 objects, 2 red, 3 blue and 5 green. The objects are indistinguishable. In how many ways can they be arranged on a line? As there are 3 groups of objects I did that: $ 10! / ...
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37 views

52-Deck 3 Cards Drawn Possible Combinations Question

I have a HW problem I'm trying to pin down and I think I'm confusing myself... Question: In a card game w/ a standard 52 card deck, a hand is a set of 3 cards. Count the # of hands that are... a) ...
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Probability Urn Group Problem

my group and I are having trouble figuring out how to do this. For some reason I have an urn that contains 10 coins. 3 of the coins are blue on one side and red on the other, 3 of the coins are blue ...
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A question on frobenius group.

I have to do this question in my assignment... I studied frobenius group from Permutation groups by Dixon and this is the given definition in it for Frobenius group so I guess I have to assume this ...
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1answer
97 views

$n!$ as a sum of $n$ positive integers

We partition $(n-1)!$ into $n-1$ parts in the following way. Consider a permutation $(a_1,a_2,\ldots,a_n)$ of $(1,2,\ldots,n)$. We say that $a_k$ dominates its predecessors if $a_j<a_k$ for ...
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1answer
31 views

What permutations of matrix entries do row and column transpositions generate?

Let $M$ be a square matrix. By transposing rows and columns, can we get any permutation of the entries of $M?$ If we can't, which permutations are generated?
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36 views

The number of bijective polynomials of particular degree in a field

I need to know please: In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d?
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4answers
118 views

Number of length $8$ binary strings with no consecutive $0$'s

How many $8$ bit strings are there with no consecutive $0$'s? I just sat an exam, and the only question I think I got wrong was the above(The decider for a high-distinction or a distinction :SSS) I ...
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1answer
29 views

C*-algebra generated by the symmetric on 3 elements

I want compute $C^*(S_3)$ where $S_3$ is the symmetric group on $\{1,2,3\}$ and $C^*(S_3)$ is the (full) C*-algebra generated by $S_3$. My attempt: Since $S_3$ is a finite group, $C^*(S_3)=C_c(S_3)$ ...
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28 views

How can i learn when to use which multiplication rule: Probability

Hey guys im studying for a math exam and was wondering if anyone has some easy techniques to remember in what kind of scenario to use these equations. These are I believe called multiplication rules. ...
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45 views

Permutation question on alphabets

Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is Well I do understand some ...
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40 views

Arithmetic of a combinations formula

I am trying to study, and I'm not quite sure how: $$ \binom{5}{3} \cdot \binom{7}{3} = 350 $$ From my understanding the formula is $$ \binom{n}{r} = \frac{n!}{r!(n-r)!} $$ Therefore: $$ ...
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25 views

Number of permutations if every element may appear in certain distance from its initial position

Suppose i have an $n-$elements array. I want to count number of permutations for which element $a_i$ allowed to appear in range $i-k, \dots, i+k,$ so $2k+1$ positions available after permutation has ...
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28 views

Difference between non - negative and positive integral solution :

Difference between non - negative and positive integral solution : (a) Number of non negative integral solution of equation $x+2y+3z+4w =n$ = Coefficient of $x^n$ in ...