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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No. of Quadrilateral having, one side, two sides, and three sides common with polygon

Number of Quadrilateral that can be made using the vertex of a polygon of $10$ sides as there vertices and having (i) Exactly $1$ sides common with the polygon (ii) Exactly $2$ sides common with the polygon (iii) Exactly $3$ sides common with the…
juantheron
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Finding the smallest positive integer $n$ such that $S_n$ contains an element of order 12

My attempt: Suppose $\sigma \in S_{n}$ such that $|\sigma|$. We know that every element of $S_{n}$ can be written as a product of disjoint cycles, that is: $\sigma = \theta_{1}... \theta_{i}$ . Now we know that the order of any element in $S_{n}$…
mizkid45
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Number of ways of arranging $n$ blue balls and $n$ red balls in a circle

I have problems reasoning out a general formula to represent the number of possible circular orientations of the $n$ red balls and $n$ blue balls as I’m very bad at combinatorial questions and when to decide what is treated as unique and what is…
Prashin Jeevaganth
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Find the number of ways of constructing $8$ using three distinct integers from {0,1,2,3,4,5,6,7,8}

In this particular example, order does matter. But at the moment, the only method I can think of is to tediously list them out. $0+1+7$ $0+2+6$ And so on. But I am thinking that there is surely a better way in general, especially when there are…
Trogdor
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The no. of ways of putting $N = p_{1}^{\alpha_{1}}.p_{2}^{\alpha_{2}}.p_{3}^{\alpha_{3}}......p_{k}^{\alpha_{k}}$ as a product of $2$ Natural no. is

The no. of ways of putting $N = p_{1}^{\alpha_{1}}.p_{2}^{\alpha_{2}}.p_{3}^{\alpha_{3}}......p_{k}^{\alpha_{k}}$ as a product of $2$ Natural no. is $\displaystyle \frac{1}{2}(\alpha_{1}+1).(\alpha_{2}+1).....(\alpha_{k}+1)\;\;,$ If $N$ is not a…
juantheron
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Showing that if all nonidentity elements have same order, the group is elementary abelian.

I can see that if $N$ is a group such that all $g \ne 1, g \in N$ have the same order, then this order is some prime $p$. Why is $N$ elementary abelian and of order $p^m$ for some $m$?
UbU
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Restricted permutation of ABRACADABRA

All letters of the word ABRACADABRA will be arranged in a row if C, R, and D are not to be together. Find how many ways are possible. My attempt number of letter A = 5 number of letter B = 2 number of letter C = 1 number of letter D = 1 number of…
Friendly Ghost
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Are all permutations of the same parity and trace of the permutation matrix the same up to the change of basis?

If two permutations $P_1$ and $P_2$ of $n$ elements have the same parity (i.e. both are even or both are odd) and the same trace of their corresponding permutation matrices (i.e. they leave the same number of elements unchanged), does this mean that…
Danijel
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Form three random groups satisfying certain parameters

I want to form three football teams out of 18 players that have been playing for years. The input data is the following: 1) skill index per player, and 2) 100+ game results, including the most successful combination of two and three players. The…
NBK
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How many double letter mutations are possible in a certain DNA string?

We are currently doing permutations and factorials in my Maths course. In this week's online quiz there's a question that goes like this: A DNA sequence can be represented as a string of the letters ACTG. Given a DNA sequence of length 26, how many…
Jans Rautenbach
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Number of sliding window in the string

I have to calculate number of sliding window in a string. If the sample string is "CHECKIT" and window size is 2 then CH HE EC CK KI IT Window size may vary for the string.
made_in_india
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Permutation of n ,not all different things, taking r at a time

Please read it once again, it's not as easy as it seems. Is there any Mathematical formula or finite approach to solve this type of problem. Eg - Find the permutation of word "MALAYALAM" taking 5 at a time. I have written a program which can solve…
RAHUL JAISWAL
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Breaking down permutations into disjoint cycles and transpositions, and finding inverse

I need some verification with regards to my understanding of permutations, disjoint cycles, transpositions, signs, permutation inverse and order: If a permutation is defined as…
Mohamad Moustafa
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Number of ways in which 4 letter words can formed by using the letters from the set-a,a,a,a,b,b,b,c,c,d

I thought of solving this question by making cases: 1) All four are same 2) 2 are same,2 are distinct 3) 3 are same ,1 is distinct 4) All are distinct My difficulty is about how to select identical numbers Like for case 2.How will I choose…
Abhinav
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combinations, how many ways are there?

how many ways are there to put 36 non-distinguishable balls in 15 distinguishable buckets? This is what I thought: suppose the balls are distinguishable. every time you want to put a ball in a bucket, you have 15 possibilities. so if you have to do…
Badshah
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