# Tagged Questions

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

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### Scheduling a Round Robin tournament - 4-way games

I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table. I'd like for each player to play with each other player exactly once ...
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### Decompose and compute the sign of $\sigma(k)=n+1-k$

Let $n\geq 2$ and $\sigma$ is permutationof $\{1,2,\ldots,n \}$ defined by : $$\sigma(k)=n+1-k$$ Decompose permutation $\sigma$ into product of disjoint transpositions and compute the sign of it ?...
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### How to prove that $\langle\{ (1,2),(1,2,\ldots,n) \}\rangle=\mathfrak{S}_n$

Let $n\geq 2, \tau=(1,2),\ c=(1,2,\ldots,n)$ two permutation of $\mathfrak{S}_n$ Prove that $$\biggl\langle\{ (1,2),(1,2,\ldots,n) \}\biggr\rangle=\mathfrak{S}_n$$ Indeed, normally i will ...
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### Does the concept of permutation make sense for a set indexed by the real numbers?

I know that the concept of permutation makes sense for sequences, which are sets indexed by the natural numbers (if the sequence is infinite) or indexed by the first $n$ natural numbers (if the ...
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### Formula for combinations number with aggregation

I've this scenario, 4 groups (as follow) with following values: Dimension A: A B Dimension B: K J L Dimension C: X Y Dimension D: F G I want to know numbers all possible combinations, ...
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### Permutation formula for lock combination

I know the basic permutation formula for k objects out of an n set. But what is the formula for determining the number of permutations where k is a range (1..m) ? What are the formula for the ...
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### Derangement combination calculation

For the traditional classic problem of derangement (https://en.wikipedia.org/wiki/Derangement), there is a formula $n! = (n-1)(!(n-1)+!(n-2))$, which calculates current results based on previous ...
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### Probability in $S_{15}$

We consider the set of permutations of the first fifteen natural numbers. What is the probability that $1$ and $2$ aren't contiguous? My attempt: Denote by $C_{12}=$ "The numbers $1,2$ are ...
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### Permutations of n numbers with no odd numbers next to each other

What is the number of $\{1, 2, \dots, n\}$ permutations, in which neither two neighbouring numbers are odd? Could somebody show me the reasoning that leads to the answer?
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### Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
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### How many bit strings of length $5$ do not have consecutive $1$'s?

How many bit strings of length 5 do not have consecutive 1's? I'm trying to think of a way to calculate how many ways we can arrange a string of length 5 starting with the first position (or index). ...
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### How to prove that $\langle\{ (1,2),(1,2,3) \}\rangle=\mathfrak{S}_3$

Prove that $\{ (1,2),(1,2,3) \}$ Generating set of a symmetric group $(\mathfrak{S}_3,\circ )$ SOlution provided by book we 've $(1,2,3)(1,2)(1,2,3)^2=(2,3)$ and $(1,2,3)^2(1,2)(1,2,3)=(1,3)$ ...
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### A boat is to be manned by 8 men,of whom 2 can only row on bow side & 1 can only row on stroke side;in how many ways can the crew be arranged?

I tried it by selecting 2 men out of 8 for bow side,and then arrange them in 2! ways.This can be done in$\binom{8}{2}$*2! ways,and the stroke side can be crewed in 6 ways.So the required no. of ways ...
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### How many 7-note musical scales are possible within the 12-note system?

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested ...
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### Recognizing Permutation of Group with different Label

Problem description: Assume, a group, $G \leq S_{26}$ , $S_{26}$ is a symmetric group. Each permutation of $G$ is labeled using $1,2,....26$ as usual. Suppose, $f$ is a function that changes label ...
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### how many ways is it possible to seat eight people at a round table …

I know this is a permutation problem, selecting two from eight. My problem is how to use this information: "must sit one seat away from each other." In how many ways is it possible to seat eight ...
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### difference of at least 4 between two numbers

In how many ways we can select two numbers from first $10$ natural numbers so that difference between them is at least four? If we select the numbers from the interval $[1,10]$, will the answer ...
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### distribution of 101 coins to three friends

In how many ways we can distribute 101 coins to three friends such that sum of the coins of two friends is more than or equal to the number of coins of third friend. my views:should I distribute 50 ...
Suppose there are $N$ (even number, positive) people. And each one person has to find one and only one partner to form a pair. There is also a roster within which everyone's name appearing in ...
So I have $5$ different cards and I am choosing $2$. I want to know the total possible permutations of choosing two cards (I know the answer is $25$ by using permutations). I need other ways to ...