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Questions tagged [permutations]

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

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Find all combinations that sum up to a specific number With Given Constrains

This is a continuation of this problem. Find all combinations that sum up to a specific number Think of following values. ...
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15 views

Quantity of same numbers in same positions of permutations and it's resort

If we resort numbers in permutations of length $n$ by their positions, ex. $12=12, 21=21$ $123=123, 132=132, 213=213, \color{red}{231=312}, \color{red}{312=231}, 321=321$ $1234=1234, 1243=1243, ...
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2answers
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Elements of $S_n$ can be written as a product of $k$-cycles.

Let $k\leq n$ be even. Prove that every element in $S_n$ can be written as a product of $k$-cycles. I really have no idea how to go about this. My initial intuition was to proceed by induction first ...
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Combinatory Problem, all the different ways to do this combinatory problem

if there are 10 differentiable boxes and you need to paint all them and dry the paint. how many different ways there are to do it? You need to paint the box to dry the paint. For example if you paint ...
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1answer
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8-element permutations of a multiset {3:0,1:1,1:3,1:5,1:8,1:9} with the restriction 0 is not allowed in left or rightmost position

I am lost in how to approach this problem due to the wording: Count the number of distinct 8-digit numbers that may be made by permuting the multi-set: $$MS:=\{0:3,1:1,3:1,5:1,8:1,9:1\}$$ ...
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Filling a determinant [on hold]

Suppose you have a 3×3 determinant with all bank spaces . You have to full it with only 2 numbers 1,-1 . How much of them is possible with a non-zero value
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2answers
27 views

how to make pairs from odd number of people one may be alone?

How Many ways the pairs can be formed from the group of size odd(like 3,5), one may be left alone? Eg: there are 5 students, so 2 pairs can be formed and one guy left alone... |Consider (a,b,c,d,e) ...
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1answer
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Points table in a tournament

There are $7$ teams in a tournament. Each team plays exactly one match against every other team. (Total of $21$ matches). Teams get $2$ points for a win, $1$ point for a draw and $0$ points for a loss....
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1answer
26 views

What is the definition of the least common multiple of elements of a group rather than their orders?

In an answer to my question here Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?, it is said that Let $G$ be a group, $a\in G$...
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1answer
22 views

Relationship between order of $(x,y)$ in $G \times G'$ and order of disjoint permutations, which both involve LCMs?

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice) Let x be an element of order r of a group G, and let y be an element of G' of order s. What is the order of $(x, y)$ ...
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Number of possible permutations of three pairs of socks, one blue, one black, and one white?

There are 3 colours of socks; blue, black and white and each colour has two pairs of socks. What are the possible permutations? Approach: align the right leg socks in a line, the number of ways of ...
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Find the possible states from the following virtual circuit

A virtual circuit spanning 6 nodes has 12 outstanding packets. Let ki, i = 1,2,3,4,5,6 denote the number of these packets at node i, and let the state of the virtual circuit be represented by the ...
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1answer
25 views

Is this definition equivalent to the definition of permutation of sequence given in the text. Justify your answer?

Q)In the text we defined permutations of a set and permutations of a sequence. Consider the following alternative definition of a sequence $\sigma : I \to A$. Let S be the set of elements that appear ...
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2answers
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How many permutations are possible of $n_1$ a's, $n_2$ b's, and $n_3$ c's, such that no two a's and no two b's are adjacent?

Given $n_1$ number of a's, $n_2$ number of b's, $n_3$ number of c's. They form a sequence using all these characters such that no two a's and no two b's are adjacent. (a and b can be adjacent, but ...
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Two dimensional sorting: finding $\max_\sigma \langle M^\sigma,C\rangle$ for permutation function $\sigma$

Suppose that we have a non-negative matrices $C,M\in \mathbb{R}^{m\times m}_{\ge0}$. Also suppose $\sigma:[m] \rightarrow [m]$ is a permutation of $\{1,2,...,m\}$. We want to find the permutation that ...
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2answers
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How many 4-letter words can be formed by the word mississippi so we have letter i in all arrangements

How many 4 letter words can be formed by the word "mississippi" so we have letter "i" in all arrangements? What I've done was to include "i" to one those 4 words, so we would have 3 words to choose ...
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2answers
39 views

For a matrix $A$ find the smallest positive $n$ for which $A^n=I$

Let $A$ be a $8 \times 8$ square matrix with $a_{12} = a_{24} = a_{33} = a_{41} = a_{58} = a_{65} = a_{77} = a_{86} = 1$ and all other entries $0$. Then $A^n=I$ for some $n$; find $n$. I successfully ...
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1answer
30 views

In probability, Is there a relation between k-permutations and the multinomial coefficient ?

If we want to place 8 rooks on an 8x8 chessboard, then the number of all the possible placements is 64!/(64-8)! which is just 64-P-8 (k-permutation) But, can't the same problem be approached as ...
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2answers
36 views

Let $A = \{1, 2, 3\}$. Write down all the permutations of $A$. Suppose $B = \{1, 2, 3, 4\}$. How many permutations of B are there?

Let $A = \{1, 2, 3\}$. Write down all the permutations of $A$. Suppose $B = \{1, 2, 3, 4\}$. How many permutations of B are there? How would you generate all permutations of $B$ in a systematic way, ...
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group of rigid motions vs. symmetry group

I'm slightly confused when it comes to terminology, I am following Judson's Abstract Algebra and came across the following: "The group of rigid motions of a square consists of 8 elements [...]" it ...
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1answer
39 views

Permutations with groups of different sizes

Hi I am unable to find any resources online about my current problem. What I have is a bowl with $x$ balls and I want to know how many orders are possible when I pull out $k$ balls. My problem is that ...
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1answer
35 views

How many permutations can there be?

Suppose I have a set $$\{1,...,1,2,...,2,3,...,3,...,n,...,n\}.$$ How many permutations can I have? I did try to solve this but I could only do it by brute force and for small $n$. I think the answer ...
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Total Possible Combinations when exhausting a group

I have 3 green basketballs, 2 red, 6 yellow and 4 green. How many combinations are possible if I shoot one basketball at a time with the stipulation that once a color is chosen, I have to shoot all of ...
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Combinations of $64\times64$ image

I have a $64\times64$ image where the pixels can have either $0$ or $1$ value. I want to divide $0$ and $1$ equally between all the pixels. Nowhere are two cases: 1) where the order of pixels don't ...
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2answers
65 views

Randomly permute $\{1,\cdots,100\}$. What is the probability that none of the $S_k$'s defined by $\sigma(1)+\cdots+\sigma(k)$ is divisible by $3$?

After randomly permuting the numbers from $1$ to $100$, what is the probability that none of the $S_k$'s defined by $S_k =\sigma(1)+\cdots+\sigma(k)$ is divisible by $3$? I think I have a ...
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2answers
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Is Partition Formula wrong when dividing 2 into 2 classes?

In basic combinatorics, when dividing a total of N objects into k classes each with $r_k$ objects we have Partition Formula: $$ \binom{N}{r_1}\binom{N-r_1}{r_2}...\binom{r_k}{r_k}=\frac{N!}{r_1! r_2!.....
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1answer
42 views

Why did this reversal from the left cosets of $\langle (1, 2, 3) \rangle$ in $A_4$ give me the right cosets?

In this question How to derive the cosets of $A_4$? I derived that the left cosets are $$\{ 1\langle (1, 2, 3) \rangle, (143)\langle (1, 2, 3) \rangle, (142)\langle (1, 2, 3) \rangle, (341)\langle (1,...
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Number of words in mumbo-jumbo tribe's language

A tribe named Mumbo-Jumbo has $3$ letters in their alphabet, how many words are there with length no greater than $4$? I answered $3^4+3^3+3^2+3^1$. The author's answer is $40$, did the author add up $...
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Number of ways to seat people in a rectangular seat arrangment

Their is an (2 X n) size seat arrangement is given and we've to make m people sit in this seat arrangement , such that no two people share a side . example n=3 m=2 their are 8 ways of seat ...
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Nearest signed permutation matrix to a given matrix $A$

Let $A \in \mathbb{R}^{n\times n}$ be a square matrix and let $Q \in O(n)$ be the nearest orthogonal matrix to $A$ under the Frobenius norm, i.e. $$Q = \text{arg}\min_{M \in O(n)} ||A - M||_{F}^2$$ ...
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1answer
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How to create subsequences from a set of ordered integers given the specified constraints.

Given, for example, the following set of integers $\{1,2,3,4\}$, how can you compute the number of all possible sequence scenarios, where a scenario consists of a number of sequences, as following ...
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4answers
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There are $3$ boys and $3$ girls. How many ways can they be arranged in a way that two girls won't be together?

There are $3$ boys and $3$ girls. How many ways can they be arranged in a way that two girls won't be together? Instead of making things harder, I directly write an assumption as illustrated below ...
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1answer
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Question about binomial distribution and permutation.

A multiple-choice test has 15 questions, each having 4 possible answers, of which only 1 is correct. If the questions are answered at random, what is the probability of getting all of them right? I ...
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3answers
29 views

A five-digit number is formed using digits 1, 3, 5, 7 and 9 without repeating any one of them. What is the sum of all such possible numbers?

I have solved the above question but I found another method for it on a website but didn't understand. Alternative method: The sum of all the numbers formed by the digits a1, a2, a3,……….an, without ...
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1answer
32 views

Not all symmetric sets are shift-invariant

In my probability course, symmetric sets are Borel-measurable sets defined as follows: $$ \begin{equation} \mathcal{S}=\{ S\in \mathcal{B}(\mathbb{R}^\mathbb{N}):p(S)=S \text{ for any finite ...
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1answer
86 views

Arranging 3 types of balls.

There are $b$ black balls, $w$ white balls and $g$ green balls. Find the number of ways of arranging them such that no two white balls are consecutive and no two green balls are consecutive. (Black ...
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Are there a formula to give index of each element after ordering permutation if we know permutation number?

Suppose we have 3 unique elements, let give each element its own index 0,1,2 It can be reorder into 3! permutation which is 6 ways to order this list. And each one of permutation can be given a ...
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0answers
27 views

Abstract Algebra Proof with cycles and transpositions

Let s be a permutation from Sn, for some n. Consider standard (unique) representation of s as a product s1s2...sk of independent cycles and let dec(s) be the decrement of s. Let (ij) be arbitrary ...
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2answers
63 views

How many 9-digit numbers are there with an even digit sum?

How many 9-digit numbers are there with an even digit sum? How do I approach this task? I know the answer, but want to see how I need to think about solving it. My approach to solving this was very ...
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1answer
24 views

Finding the expectation of a random variable on $S_n$

We have been given a random permutation on $n$ letters. The random variable $X$ is equal to the number of times that the position of an element after permutation is larger than its value. Find $\...
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1answer
36 views

Alternative proof that the 3-cycles generate the alternating group $A_n$

A lot of proofs I have seen involve writing $p \in A_n$ as a product of an even number of transpositions. I have a different proof that involves disjointedness like in my question earlier today Prove $...
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1answer
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Randomly selecting and then removing vs Selecting a random permutation

I have been given an assignment for my Randomized Algorithms class. We begin with a graph and start removing vertices step by step. On each step we randomly select 1 vertix and remove that vertix plus ...
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How many four-digit positive integers with distinct digits are there in which the sum of the first two digits equals the sum of the last two digits? [on hold]

How many four-digit positive integers with distinct digits are there in which the sum of the first two digits equals the sum of the last two digits?
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3answers
62 views

Prove $(\rho_1 \rho_2 \cdots \rho_r)^u = e \implies e=\rho_1^u=\rho_2^u=\cdots=\rho_r^u$

In the proofwiki page: Order of Product of Disjoint Permutations We have a product of disjoint permutations $\pi = \rho_1 \rho_2 \cdots \rho_r$. Why exactly is it that if $\pi^u=e$, then for each $\...
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1answer
17 views

Is the identity permutation a transposition?

I've heard every permutation can be expressed as a composition of transpositions. How would this be done with the permutation $\pi:\{1\}\to\{1\}$? This only seems possible if one includes the identity ...
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2answers
150 views

Combinatorics with arrangements of the word UNIVERSALLY

How many ways are there to arrange the letters in UNIVERSALLY so that the four vowels appear in two cluster of two consecutive letters with at least 2 consonants between the two clusters? For this, ...
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4answers
48 views

Why do different permutations $pq$ and $qp$ have the same length?

Does the reverse composition (or reverse multiplication) of permutations have the same cycle length? Let $p$ and $q$ be elements $S_5$: $q = (1 4 5 2)$, $p = (5 2) (1 3 4)$. Permutation multiplication ...
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0answers
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How many binary strings of length $10$ contain at least $4$ $0$'s?

I'm not sure how do deal with the at least part. The way I understand it is that I have to the opposite, so find all the number of ways that there's no zero, $1$ zero, $2$ zeros, and $3$ zeros. ...
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1answer
32 views

Describe the subgroups of S_5 generated by the 5-cycles

I'm new to Group theory and I'm just checking on my understanding. One example of 5-cycle is $(1\ 2\ 3\ 4\ 5)$. Hence, a subgroup generated by this 5-cycle consist of $\{(1\ 2\ 3\ 4\ 5), (1\ 3\ 5\ 2\ ...
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1answer
145 views

The house numbers on a street each have 4 digits that range from 1000 to 9999. How many of these house numbers have exactly 2 digits that are 5?

My problem with this question is the fact that the first number doesn't have as many choices as the last 3. I'm sure it's easier than I thing it is but I just can't get around it. I have 4C2 ways of ...