Questions tagged [permutations]

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

The word permutation has several possible meanings, based on context. In combinatorics, a permutation is generally taken to be a sequence containing every element from a finite set exactly once. Permutations of a finite set can be thought of as exactly the ways in which the elements of the set can be ordered.

In group theory, a permutation of a (not necessarily finite) set $S$ is a bijection $\sigma : S \to S$. The set of all permutations of $S$ forms a group under composition, called the symmetric group on $S$.

Reference: Permutation.

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How to prove product of even permutations is even

How to prove product of even permutations is even? I've found if the even permutations are disjoint, then it's easy because an even number times another even number is even...so the product of two disjoint even permutations is even...but how about…
Ian
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number of possible password combinations

There is a computer password , whose restriction is that it . 1)Each character is an upper case alphabet (A...Z) or a digit (0 to 9)) 2) It should be of length 6 3) It should have at least 1 digit I solved it as combinations of password with at…
Harish Kayarohanam
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inclusion and exclusion practice exercise

This is a practice question. I understand Inclusion and Exclusion but I am having a terrible time setting questions up correctly. This is on of the practice questions in the text. At a flower shop they want to arrange $15$ different plants on five…
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How To Count Shuffle permutations

Let $n\in \mathbb{N}$ and $S(n)$ the permutation group on $\{1,\ldots,n\}$. For any $p,q\in \mathbb{N}$ with $p+q=n$, the set $Sh(p,q)\subset S(n)$ is the set of all permutations $\tau$ such that $\tau(1)< \cdots \tau(p)$ and $\tau(p+1)<\cdots…
Mirco
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How do I know if I should select simultaneously or separately?

When using combinations to calculate how many ways an event occurs, how do I know whether to select objects simultaneously or separately? For example, I am given the task of finding the probability of rolling a two-pair in poker dice. A two-pair is…
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Possible combinations of letters

I have an example in my lecture where I have the letters: A,A,A,B,B,C,C,C and I need to give the possible combinations. $$ \frac{8!}{3!\cdot3!\cdot2!} = 560 $$ In one part of the exercise I have to give the number of arrangements where 3 A's are…
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number of permutations maximizing a sum

Let $n$ be an odd integer greater than $1$. Find the number of permutations $\sigma$ of the set $\{1,\cdots, n\}$ for which $|\sigma(1) - 1| + |\sigma(2) - 2|+\cdots + |\sigma(n) - n| = \frac{n^2 - 1}2$. I found the solution below from a book, but…
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Permutations of $[n]$ where no 3 numbers congruent $\mod k$ are consecutive

Question: Given natural numbers $k$ and $n$, in how many ways can we order the numbers $1,2,\dots, n$ such that no three consecutive numbers are congruent $\mod k$? I was reading this paper where the problem for no adjacent pairs is solved and…
ploosu2
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Question from isi previous years

(a) Show that $\left(\begin{array}{l}n \\ k\end{array}\right)=\sum_{m=k}^{n}\left(\begin{array}{c}m-1 \\ k-1\end{array}\right)$. (b) Prove that $$ \left(\begin{array}{l} n \\ 1 \end{array}\right)-\frac{1}{2}\left(\begin{array}{l} n…
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Is a transitive group more likely to be soluble if its degree has more prime divisors?

There is some empirical evidence that the probability that a permutation group of a given degree is soluble increases with the number of prime divisors of the degree. (Primes are counted with multiplicity so, for instance, $12$ has $3$ prime…
James
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How many ways nine people can sit on these chairs if at least one of the chairs in row two must remain unoccupied?

Question: find how many ways nine people can sit on these chairs if at least one of the chairs in row two must remain unoccupied. Attempt: $9!+(9!)(10C9)(6)+(9!)(11C9)(3)=82010880$ Reasoning: if three chairs are unoccupied, then there are $9!$…
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Every action gives rise to a homomorphism

I'm reading Chapter 3 from the book "The theory of finite groups" by Kurzweil and Stellmacher where they say the following: Let $G$ act on a set $\Omega$. That is, for each $x\in G$ and $\alpha\in\Omega$, there exists an element $\alpha^x\in \Omega$…
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How many quartic polynomials have single-digit integer coefficients?

Let $X$ be the set of all polynomials of degree 4 in a single variable $t$ such that every coefficient is a single-digit nonnegative integer. Find the cardinality of $X$. This is a question from Balakrishnan's Intro. Discrete Mathematics, and…
Jonathan Dewein
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Enumerating the square equivalent classes

Fill the $n\times n$ lattice square with natural numbers $\{1,2,\cdots,n\}$ each of which is used exactly $n$ times. Two configurations are in an equivalence class if and only if one is transformed from the other by any of the following group…
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Probability that in a line of 'n' distinguishable people, at least 'b' of the front 'a' spots are one of 'k' red-shirt-wearers

I've been approaching this problem for some time. Imagine that we have n distinguishable (uniquely named) people lined up in a line where k of these people are wearing red shirts. I want to find the probability that at least b of the front a spots…
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