Tagged Questions

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

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Covering pairs with permutations

Consider an $n \times n$ matrix $M_n$ with the following properties: Each row is a permutation of $A_n \equiv \{1, 2, ..., n\}$. Every ordered pair $(i,j)$, $i,j \in A_n$, $i \neq j$, appears as a ...
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For any permutation $\sigma \in S_n$, $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even when $n$ is odd [closed]

Let σ be a permutation of ${1, 2, 3, . . . , n}$, n odd. I want to show that $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even. Thank you.
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Numbers of words allowing repetition [duplicate]

10 different letters of an alphabet are given. words with 5 letters are formed from these given letters.I have to determine the number of words which have at least one letter repeated. Answer is - ...
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sum of numbers formed by permutations

I have digits 2,3,4,5. I have been asked to find the sum of all 4 digits the numbers that can be formed using these digits without repetition such that all are included in the number. Can someone ...
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I need help answering a few simple math problems related to permutations and probability

Question 1: How many words can you make from the letters Texas if repeats are not allowed? Question 2: How many words can you make from the letters Texas if repeats are allowed? Question 3: What is ...
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Combinatorics problems involving permutations

Let $A= \{ 1,2,3,...,n\}$ a set and $f:A \to A$ a permutation of the set A. We call a number $x \in \{ 2,3,...,n-1 \}$ special if $f(x)>\max \{f(x-1),f(x+1) \}$ or $f(x)<\min \{f(x-1),f(x+1) \}.$...
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How many possible placements are there for a Battleship puzzle?

I am studying the NP-Completeness of the battleship puzzle; the pencil and paper game found in newspapers and not the more popular 2-player version. I understand why the puzzle is NP-Complete because ...
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How many straight lines can be made between 10 points such that 4 of them are colinear?

So i know how to get the answer. We just have to find $C(10,2)$ and subtract $C(4,2)$ and add 1. We are basically counting all the points between co-linear points as 1. So the question is why we are ...
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How many mixed double pairs can be made from 7 married couples provided that no husband and wife plays in a same set?

So for first man there can be 7 possible partners including his wife, for the next man there will be 6 possible partners and so on, $therefore$ for $7$ men and $7$ women, there will be $7!$ possible ...
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Finding number of functions from a set to itself such that $f(f(x)) = x$

The questions states that $f: A\rightarrow A$ is a function which satisfies $f(f(x)) = x.$ We have to find the number of such functions with $A = \left\{1,2,3,4\right\}$. The given condition clearly ...
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A question of permutations and combinations with six cards and six envelopes.

Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same ...
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How do I calculate the number of unique permutations in a list with repeated elements? [duplicate]

I know that I can get the number of permutations of items in a list without repetition using (n!) How would I calculate the number of unique permutations when a ...
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Painting a 2x2 Grid

We have a 2x2 grid and 10 different colours. I want to paint such that adjacent grids are painted with different colors. How many ways can i do this? ...
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$5$ chem students, $6$ maths students and $7$ physics students permutation

$5$ chem students, $6$ maths students and $7$ physics students. Find the number of arrangements if a)Chem majors are to occupy the first 5 positions b)Chem majors cannot occupy the first 5 positions ...
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Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ (...
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Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
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If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
Prove $sgn(π) = sgn(π^{-1})$?
I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...