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

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2answers
76 views

How many 20 digit numbers have 10 even and 10 odd digits?

How can I perform operations so as to get this value? Number should not have leading zeros.
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1answer
19 views

For $f\in\mathbb{Q}[x]$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$

Let $f\in \mathbb{Q}[x]$ a monic irreducible polynomial, and Gal($f$) be a subgroup of $S_n$. How do I prove that Gal($f$) $\subset A_n\iff \Delta(f)$ is a square in $\mathbb{Q}^*$? I know what ...
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4answers
48 views

How many numbers are possible from $a^x b^y c^z$?

How to calculate total nos of possible value made from given numbers. e.g. : $2^2 \cdot 3^1 \cdot 5^1$ . There $2$ , $3$ , $5$ , $2\cdot2$ , $2\cdot3$ , $2\cdot5$ , $3\cdot5$ , $2\cdot2\cdot3$ , ...
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2answers
34 views

can anybody help in finding number of ways the letters of the word 'PERMUTATION' be arranged so that consonants are in alphabetical order? [closed]

I had tried the question and got the answer 11!/(6!2!) but the answer given is 11!/6! if any body can explain that why 2! is not in the answer or the answer is wrong.
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0answers
25 views

What happens to the Permutation Rule when r=0?

This is small but quirky idea that popped into my head in the middle of the night last night. If I have $n$ objects, and want to find out how many permutations (sequences) of $r$ objects there are, ...
3
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2answers
42 views

Unable to derive reason/formula for permutation problem

What is the probability of $n$ preceding $1$ and $n$ preceding $2$ when we randomly select a permutation of ${1, 2, . . . , n}$ where $n ≥ 4$? I wrote out examples of n! when n equals some number ...
2
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2answers
83 views

German combinatoric terms vs English terms

I'm a German Computer Science student and I currently work with combinatorics as part of my curriculum. I wanted to research combinatorics in English but I'm confused about the terminology. In German ...
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2answers
34 views

increasing, decreasing, non-decreasing, non-increasing permutation/combination

I have this question and I'm stuck Q: In the set of three-digit integers {100,101,...,999}, how many integers are there (a) with three distinct digits that are either increasing (as in 257, 139) or ...
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2answers
36 views

show that A(T) is a group under operation of composition of functions

Problem: Let T be a nonempty set and A(T) the set of all permuations of T. Show that A(T) is a group under the operation of composition of functions. Permutation of the set T is the bijective ...
4
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1answer
58 views

There are $5$ apples $10$ mangoes and $15$ oranges in a basket.

There are $5$ apples $10$ mangoes and $15$ oranges in a basket. Then find number of ways of distributing $15$ fruits each to $2$ persons. Can I approach this question as number of ways $15$ fruits ...
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2answers
84 views

How many combination of $3$ integers reach given number?

I have 3 numbers $M=10$ $N=5$ $I=2$ Suppose I have been given number $R$ as input that is equal to $40$ so in how many ways these $3$ numbers arrange them selves to reach $40$ e.g. ...
2
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0answers
20 views

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$ [duplicate]

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$. I'm not quite sure how to start this. The solution starts ...
1
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1answer
32 views

number of possible subsets formed with odd count of odd numbers = $2^{n_{odd}-1} \cdot 2^{ n_{even}}$

Let $n_{odd}$ represents the number of odd numbers in the set $S$ and $n_{even}$ denote the number of even numbers. The total number of possible subsets formed with odd count of odd numbers ...
1
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2answers
68 views

Prove that there is no permutation $\sigma$ such that $\sigma(123)\sigma^{-1}=(124)(567)$

Prove that there is no permutation $\sigma$ such that $\sigma(123)\sigma^{-1}=(124)(567)$. Cycle $(123),(124),$ and $(567)$ has order $3$ so if the equation $\sigma(123)\sigma^{-1}=(124)(567)$ is ...
0
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2answers
30 views

Number of ways so that at least one soldier find that soldier next to him is also selected.

20 soldiers are standing in a row and their captain want to send 7 out of them for a mission. In how many ways can captain select them such that at least one soldier find that soldier next to him is ...
4
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2answers
72 views

$15$ men and $15$ women into $15$ couples

Find the number of ways of dividing $15$ men and $15$ women into $15$ couples. My solution is: First Man can be paired with any one of $15$ women in $15$ ways. Second Man can be paired in $14$ ...
1
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2answers
95 views

How many ways are there to assign $20$ different people to $3$ different rooms with at least $1$ person in each room?

How many ways are there to assign $20$ different people to $3$ different rooms with at least $1$ person in each room? I know how to approach this problem using combinations:$$17!\cdot {3 \choose ...
0
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0answers
8 views

presentation of the symmetric group via transpositions fixing one element

Consider the symmetric group $S_n$. If we use the most popular set of generators $\sigma_1, \sigma_2,\cdots,\sigma_{n-1}$ with $sigma_i$ being the transposition $(i \, i+1)$, it is well known that ...
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0answers
27 views

Divison in factoradic base

I'm trying to find am means of dividing two numbers in factoradic base. So far goolging seems to turn up nothing at all. Is there a better way of doing this than long-division? I'm hoping for ...
2
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1answer
35 views

system of two eqautions in three unknowns: finding the number of solutions

I have a system of two equations with three unknowns. $$x+y+z=7$$ $$x+2y+3z=10$$ On solving, I got the following values. $$ y = a$$ $$x = (11-a)/2$$ $$z = (3-a)/2$$ How would I go about finding ...
1
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1answer
37 views

Finding number of possible hands in 5 card stud when order is matter

I'm trying to determine the game 5 card poker when order is matter all my trails ended with fail except the one pair. The way I did one pair assuming it is $\{k,k,x,y,z\}$ $^{13}P_1 \times ^4P_2 ...
0
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1answer
66 views

Number of ways to arrange identical coins in a ring [duplicate]

I have $10$ identical coins that are to be placed at $10$ sites arranged in a ring. Assuming that the coins are placed at random, and by some chance only three coins turned out to be "up-faced". ...
1
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1answer
20 views

Give in cyclic form an odd permutation $\pi$ so that…

Give in cyclic form an odd permutation $\pi$ so that $\pi ∈ S_9$ so that $\pi^2 \neq id$ and $\pi^{10} = id$ I have started like this: $o(\pi) | 10$ which means that $o(\pi) = 1,2,5,10$. I don't ...
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3answers
31 views

Find a matrix that results in a permutation

Apologies for the sort of vague title, but part of my problem is that I'm not quite sure of what my problem actually is asking! Given a vector $v_n = \begin{pmatrix} 1\\ ... \\n \end{pmatrix} $ , ...
1
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1answer
56 views

How many ordered subsets of a set?

We have a set $A$ consisting of $n$ elements. Is there a closed form for the total number of subsets when you care about the order of the elements in the subsets? Lets call the number ...
1
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1answer
61 views

Product of permutation matrices is the matrix of the composition

I want to prove that the product of two permutation matrices equals the matrix of the composition of the two permutations: $M(\sigma).M(\tau)=M(\sigma \circ \tau)$, where both $M(\sigma)$ and ...
1
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4answers
58 views

Eight digit number is formed using all the digits: $1,1,2,2,3,3,4,5$

Eight digit number is formed using all the digits: $1,1,2,2,3,3,4,5$. Find Total numbers in which no two identical digits appear together. Exactly two pair of identical digits occur together. For ...
0
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1answer
34 views

Number of permutations with limited number of repetitions

How many four-digit numbers can be formed with the numbers 1, 1, 1, 2, 2, 3, 3, 4 (it means you can use 1 for three times, 2 and 3 two times, and 4 just once)? How does a generalization of that look ...
3
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1answer
38 views

Am I interpreting this subgroup of the permutations of $n$ elements $S_n$ correctly?

I am asked to prove $H=\{\sigma \in S_n:n\sigma=n\}$ is a subgroup of $S_n$. Am I correct in thinking this is the set of all permutations that map any element from $\{1,2,...,n\}$ to itself? That is ...
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0answers
29 views

whole number solution using permutation and combination

I am trying to solve this question. What I have understood is that we can solve the problem using the following method.Suppose we have to divide $x$ CD's among $4$ nephews answer will be simply Number ...
1
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1answer
55 views

Compute the character of $\pi$ and decompose into irreducible representations

$V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ $\mathbb{C}$ has standard basis $e_1, e_2$ and V has basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ $\pi$ is a representation of ...
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2answers
22 views

Maximum number of intersections problem

We are given 'n' different lines and 'm' different circles. What is the maximum number of intersection points using these lines and circles? I have no idea how to approach this. Help?
1
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1answer
19 views

Permutation of rows with repetition

A binary matrix with exactly one entry of 1 in each row and 0s elsewhere performs a "permutation with repetition" of the rows of the matrix it left-multiplies. Example: $$ \begin{bmatrix} 1 & 0 ...
0
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0answers
41 views

Show that the symmetries of a graph are a permutation group

A Symmetry of a graph X is a permutation of the vertices that also permutes the edges. The distances between vertices are preserved. Show that the symmetries of a graph are a permutation group, So ...
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0answers
30 views

Permutation of cycles as transposition

Every permutation of a finite set can be written as a cycle or as a product of disjoint cycles Every permutation in $S_n \forall n \geq$ 2 can be written as a product of transposition. ...
3
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2answers
43 views

If a subgroup of $G$ can be described as a set of permutations with certain cycle types, is it normal in $G$?

I know that we can show $A_n \trianglelefteq S_n$ by observing that: $A_n$ is the set of all permutations in $S_n$ which are a product of an even number of transpositions. Hence any element is the ...
0
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0answers
29 views

Minimum gems required to make a garland containing all permutations, with uniqe colored gems, of size n, where we have infinite gems of N colors.

What is the minimum number of gems required to make a garland (circular) which contains all permutations, with unique colored gems considered as a valid permutation, of size n, when we have infinite ...
0
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1answer
35 views

Identity element as product of transposition.

My lecture notes indicates that the identity element of a symmetric group $S_{n}$ is $\left ( \right )=\left ( 12 \right )\left ( 12 \right )$ Just to state: The identity element $b \in G$ ...
0
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1answer
23 views

Total possible combinations of variables

I have 4 factors, each containing a different number of entry. For example: I would like to list and compute the number of possible combinations. So each combination consist of 1 fruit, 1 drink, 1 ...
0
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1answer
25 views

RGB color combinations

RGB colors are selected by 3 selectors: Red, Green, and Blue. Each of these can be between $0$ and $255$. So (and I'm sure this is some kind of permutation but I can't put my finger on the actual ...
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1answer
25 views

Understanding proof for order of permutation

Theorem:The order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles. The proof for the order of permutation as extracted ...
2
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1answer
54 views

Permutation with constrained repetition: Distribution of random variable “number of pairs of identical elements”

You have a string of 360 letters: 180 x 'A' and 180 x 'B'. The number of ways this string can be permuted is $$\frac{360!}{180!180!} = \binom{360}{180}.$$ Assume the permutation is constricted ...
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4answers
35 views

There are $5$ women, $3$ men. How many ways to form a committee of $3$ with at least $1$ member of the opposite sex?

I have looked through several topics for similar solutions and I have attempted an answer to the question. Unfortunately, the sample question itself does not have an answer. From $5$ women and $3$ ...
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1answer
49 views

Permutations with limited repetition and various constraints

you have a string of 360 letters: 180 x 'A' and 180 x 'B'. I (hope I) understand that the number of ways this string can be permuted is $$\frac{360!}{180!180!} = \binom{360}{180}$$ What I have ...
0
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1answer
17 views

problem with alternating group with order 3

I was doing some computation with $A_3$, the alternating group on 3 letters. I know it has to be abelian, even cyclic, but when I carried out the actual computation... I couldn't make sense out of it. ...
0
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1answer
30 views

Product of disjoint cycles

I am working on a problem and it ask to solve for the product of disjoint cycle from left to right convention. Have I done anything wrong in my attempt? $$(16527348)\cdot (152468)\cdot( 37 )$$ ...
0
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0answers
22 views

Permutation of vertices of a square $(D_{4})$

$D_{4}$, the dihedral group of order 8 is a square. Suppose the vertices are labelled in an anti-clockwise fashion, starting with 1 at the bottom right corner. Each rotation is $\frac{2\pi}{4}$=90 ...
2
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4answers
77 views

selecting 4 non-consecutive books from 10 books.

I have a set a $10$ book kept in a line and I want to find out how many ways $4$ books can be chosen from that if I don't choose consecutive books to be taken out. I felt this is similar to ...
2
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1answer
15 views

Rolling Dice Probability

A fair dice is rolled 3 times, The probability of the product of the three outcomes is a prime number is? The products which give a prime number I found out to be only 4. However for the total ...
4
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2answers
98 views

The set of all even permutations in G forms a subgroup of G

Show that if $G$ is any group of permutations, then the set of all even permutations in $G$ forms a subgroup of $G$. I know that I need to show the closure, identity, and inverses properties ...