Mathematics Stack Exchange
Mathematics Stack Exchange

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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In how many ways the letters of the word RAINBOW be arranged such that A is always before I and I is always before O.

I research some sites and books and i found these this approach helpful but could not understand a bit. Approach: All the 7 letters of the word can be arranged in 7! ways. and 3particular letters can be arranged in 3! ways..But the given condition…
Jack
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Missionary and Cannibal problem

I got an interesting problem yesterday (Yes, for homework, but it seems like this is on topic) The problem goes like this: Three missionaries and three cannibals wish to cross a river. There is a boat which can carry three people and either a…
Nico A
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Conjugation of permutation group $S_n$

I want to find the conjugacy classes of the permutation group $S_n$ To start with I think I have to prove that $\pi(\sigma_1\dots \sigma_m)\pi^{-1} = (\pi(\sigma_1)\dots \pi(\sigma_m))$. Where $\pi$ is an arbitrary element and $\sigma$ is a simple…
john
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Number of permutations of an integer

If $n$ is an integer, how many permutations are less than, equal to and greater than $n$? For example if $n=24335$, $43325\gt n$, $23345\lt n$, etc...
JMP
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Commutativity of cycles

Disjoint cycles commute: $(ab)(cd) = (cd)(ab)$, but do non-disjoint cycles commute? Does $(ac)(ab) = (ab)(ac)?$ Consider the composition of two permutations: $\begin{pmatrix} a & c\\ c & a \\ …
user255751
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Number of permutations in a word ignoring the consecutive repeated characters

Given a word "aab", permutations are: aab, aab, aba, aba, baa, baa I need to get the number of permutations where characters don't repeat. So from the above permutations, I need to ignore those which has consecutive characters. So, I need to the…
Placid
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Trying to learn how to compose permutations

I am trying to prove myself that $(1)(2)(3)(4) = (12)(12)(3)(4).$ So, $\begin{pmatrix} 1 & 2 \\ 2 & 1 \\ \end{pmatrix}$ $\begin{pmatrix} 1 & 2 \\ 2 & 1 \\ …
user255751
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permutations and probabilty

In a certain country, the number plate on a car consists of any 3 letters of the alphabet (the first letter is always a "K" or a "G"), followed by any 3 digits (0 to 9) and a alphabet. For a car chosen at random, what is the probability that the…
laki
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Find The number of ways?

The number of ways in which all the integers from 1 to 36 (both inclusive) can be arranged such that no two multiples of 6 are adjacent is expressed as $$ m! x^n Pr $$ where m, n, r are distinct positive integers. What is the sum m + n + r? How…
vikiiii
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How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed.

Q. How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed. A. $21$ Answer is correct (please provide a thorough explanation). Unit digit nos. : $3$ Dual digit nos. : $2×3$ Three digit nos. : $2×2×3$…
Kountay Dwivedi
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The number of possible combination of column values with possibly common elements

I would like to calculate possible combinations for a given set of data: There is an x amount of columns (let's say 3) each column contains y amount of words (lets say 2), now I would like to calculate total amount of permutations possible, the…
Samadi
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Even permutations

I am given the symmetric group $S_{9}.$ Let $$\sigma = \begin{bmatrix} 1 & 2& 3& 4& 5& 6& 7&8 &9 \\ 4& 8& 7& 9& 3& 1& 2& 5 & 6 \end{bmatrix}$$ Now the question asks: Is $\sigma \in A_{9},$ the subgroup of even permutation on $1,2,…
Mathematicing
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Showing the permutation of 2 elements in a symmetry group is an even permutation

Show that for every 2 elements $\alpha$ and $\beta$ in $S_{8}$, the permutation $\alpha ^{-1}\beta ^{2}\alpha $ is an even permutation. How do I show that the above is an even permutation? I know that permutations are always either even or dd and…
sourcecode
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When to close brackets in product of disjoint cycle

when expressing 2 composition of function as a product of disjoint cycles, when do we 'close' the bracket? None of the sources explain this clearly. Some do not even make an attempt to.
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