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

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1answer
36 views

Is an orthogonal matrix necessarily a permutation matrix?

Is an orthogonal matrix necessarily a permutation matrix? I believe the answer is no as a permutation matrix is a special case of an orthogonal matrix, but I am having a trouble finding a ...
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1answer
14 views

Find the left coset $(13)H$ of $H= \langle (12) \rangle$ in $S_5$

I'm a bit stuck on this problem. I understand that $S_5$ is the group of symmetries with $n=5$, that's trivial. I'm also aware that there are $120$ subgroups in $S_5$ and $2$ in $H=\langle(12)\rangle$ ...
0
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1answer
41 views

Permutation of boy and girl names

During your pregnancy, you decided on a list of 23 girls’ first names and 16 boys’ first names, as well as a list of 11 gender-neutral middle names. To your surprise, you had quintuplets, two boys and ...
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0answers
23 views

“Tetris permutation” set generation

In the game of tetris you are guaranteed to get each of the 7 unique pieces in some random order. For this we will call them abcdefg. This would give us ...
2
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1answer
32 views

order of $(1,3,2,4)(4,3,5)$

Write the $ \pi=(1,3,2,4)(4,3,5) $ as a) product of disjoint cycles b) product of transpositions, and is $\pi$ odd or even? c) order of $\pi$ I think this is right $$(1,3,2,4)(4,3,5)= \begin{...
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0answers
35 views

Is it true in the Group of Symmetries that (ab)(ac) = (acb)?

Suppose $1 \le a,b,c \le n$ and $a\ne b$, $a\ne c$, $b\ne c$. Is it true in $S_n$ that $(ab)(ac)=(acb)$? I'm generally new to the subject of Applied/Abstract Algebra and feel as if this is easier ...
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2answers
34 views

Can you create a formula for the amount of possible permutations of a three digit number, who has a digit sum equal to 4

Can you create a formula for the amount of possible permutations of a three digit number, who has a digit sum equal to n?
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0answers
22 views

number of permutation in a boolean expression containing only ANDs and ORs

I need to find the number of permutations of some expression which contains only conjunctions and disjunctions e.g.: $$ e = x_1x_2 \vee x_3x_4 $$ where $x_1x_2$ and $x_3x_4$ are boolean summands, ...
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3answers
67 views

Number of subsets from an ordered set where adjacent elements may or may not be tied together

Assume we have an ordered set $S$ with a finite number of elements $S=\{1,2,3,\ldots,N\}$. I need to know the number of subsets where adjacent elements from the original set may either be tied ...
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0answers
25 views

Some properties of finite group of order $p^aq^b$

Let $G$ be a finite group of order $p^aq^b$ ( $p$, $q$ are two distinct primes and $a, b\geq 1$) with $\operatorname{Z}(G)=1$ and $P\in \operatorname{Syl}_p(G)$, $Q\in \operatorname{Syl}_q(G)$. Also ...
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1answer
52 views

Natural numbers in a circle, combinatorics, existence

I need help with a problem whose solution I'm unaware of. The first $74$ natural numbers are arrange in some manner in a circle. Does there exist an arrangement such that every sum of three ...
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1answer
46 views

Permutation question (Known answer, don't understand how to get to it)

To save your time, I simplify the question to something like this: There are $18$ people, $4$ of which are teachers. All of them($18$) are going to stand in a row. In how many ways can they be ...
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3answers
59 views

In how many ways can a positive integer $n$ be expressed as a summation of positive integers less than $n$

For example if I take $n=5$, then I can express it in the following ways: $1+1+1+1+1$ $2+3$ $3+2$ $1+4$ $4+1$ $1+1+3$ $1+3+1$ $3+1+1$ $2+2+1$ $2+1+2$ $1+2+2$ Please note that the order of terms in ...
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0answers
37 views

Find the string corresponding to a particular lexicographical rank

I came across this question few days back- Given a string - “ thereanswerisyetinsufficientmeaningfulasforadata “ , form all the words with atmost 15 letters and arrange them in lexicographical order....
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1answer
24 views

Series that converge to every real number via permutation

This great answer at MathOverflow, http://mathoverflow.net/a/29488/8784, shows that the set of permutations of $\mathbb N$ is uncountable. However, I did not grasp the fact that he uses: any ...
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0answers
12 views

Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \bigoplus_J \textbf{r}_J $$ where $\textbf{r}_J$ ...
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1answer
36 views

How many partitions of $n$ are there?

Considering a partition to be an ordered $n$-tuple $(m_1, m_2, m_3, ..., m_n)$ with all the numbers $m_i$ natural, $m_1 \le m_2 \le m_3 \le ... \le m_n$, and $m_1+m_2+...+m_n=n$: how many of those $n$-...
2
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1answer
61 views

Combinatorial proof of an identity between restricted counts of permutations and derangements

In an answer to Counting permutations with given condition, I showed that the number of permutations of $k$ elements that satisfy $\sigma(i+1)\ne\sigma(i)+1$ is $\frac{!(k+1)}k$, which is the number ...
2
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1answer
97 views

Circle Permutation w/ Restrictions questions

I was working on two permutation questions, and wasn't sure if I arrived at the correct answer. 1.) In how many ways can a family of four (mother, father, and two children) be seated at a round ...
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1answer
30 views

Permutations Question Thinking Question (8 students in circle)

For a game of London Bridge, 8 kindergarten students form a circle holding hands and then walk in a clockwise direction. If the Prefect in charge allows the children to stand wherever they wish, in ...
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2answers
156 views

Seeking combinatorial or group theoretic proof for permutation identity

While working on another problem, I found the following combinatorial equality, but I got it analytically, and I'm curious to find a counting argument. Fix $n$ a positive integer. For $n_1\leq n_2\...
2
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1answer
95 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
2
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2answers
41 views

Sitting n families around a circular table with a condition

How many ways are there for sitting n families around a circular table. Each family is a mother a father and a child. Condition: The mother and father of each family should be sitting next to each ...
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1answer
26 views

Find the number of paths given the probabilities of each move (Probability and Permutation)

A robot is programmed to move on a flat surface one step at a time, either upward (U) or downward (D) or to the left (L) or to the right (R). Each move is independent of the preceding move and the ...
2
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1answer
54 views

Meaning of 'there are exactly $2$ letters between any $2$ 'E'' (Permutation and Combination)

In a game show, the host gives an incorrect arrangement of the letters of the word 'EXCELLENT' and lets the contestant guess the word within a given time limit. Find the number of different ways that ...
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0answers
46 views

Pairs of Numbers such that the sum of their digits is Equal

How many pairs of numbers $(n,m)$ whose digits add up to the same sum, where $n\ne m$ and $(n,m)=(m,n)$ such that $m,n\le k$ , are there for a given $k$? Observing this in base 10 we are looking at ...
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3answers
96 views

In how many ways can the four walls of a room be painted with three colours so that no two adjacent walls have the same colour?

In how many ways can the four walls of a room be painted with three colours so that no two adjacent walls have the same colour ? I specifically want to use inclusion exclusion principle. So ...
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3answers
59 views

Is it permutations or combinations to find the number of ways 36 characters can be arranged in a 4-letter sequence?

If I have a number in base $36$ (a to z, 0 to 9), and I want to see how many ways those characters can be arranged in a $4$-digit number, what is that called? Is it a permutation or a combination? ...
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1answer
45 views

Permutation matrix homomorphism

Can someone please help me prove that permutation matrix is homomorphism? By that, I mean, let $f: S_n \to GL_n (\Bbb R), f(\sigma)=A_\sigma$ is homomorphism. The book tells me to prove it myself I ...
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1answer
179 views

Minimum number of balls to choose such that $k$ balls are of same color

A bag contains $a$ red balls, $b$ green balls and $c$ blue balls. We can take balls out of bag without knowing which one we choose (blindfolded). We do not replace the balls back in bag, we simply ...
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0answers
27 views

Transitive action of a $p$-group on minimal block systems

I have trouble proving the following theorem: Let $P$ be a transitive $p$-subgroup of ${\rm Sym}(A)$ with $|A| > 1$. Then any minimal $P$-block system consists of exactly $p$ blocks. Furthermore, ...
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2answers
45 views

Find how many different circular bracelets can be formed using $6n$ blue and $3$ red beads, where $n$ is a positive integer.

Find how many different circular bracelets can be formed using $6n$ blue and $3$ red beads, where $n$ is a positive integer. As these are circular permutations where flipping does not make any ...
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0answers
38 views

Product of Cycles: Name to denote “direction” of composition

Is there a notation to denote the difference between these two products of cycles? It seems as though there are two conventions out there that should have a specific name for them. The subscripts for ...
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3answers
73 views

Flag Permutations problem

Hi I'm trying to understand Permutations and Combinations in depth and I have this question: How many ways are there to place $25$ different flags on $10$ numbered (diff) flagpoles if the order of ...
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2answers
95 views

How many 20 digit numbers have 10 even and 10 odd digits?

How can I perform operations so as to get this value? Number should not have leading zeros.
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1answer
20 views

For $f\in\mathbb{Q}[x]$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$

Let $f\in \mathbb{Q}[x]$ a monic irreducible polynomial, and Gal($f$) be a subgroup of $S_n$. How do I prove that Gal($f$) $\subset A_n\iff \Delta(f)$ is a square in $\mathbb{Q}^*$? I know what ...
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4answers
52 views

How many numbers are possible from $a^x b^y c^z$?

How to calculate total nos of possible value made from given numbers. e.g. : $2^2 \cdot 3^1 \cdot 5^1$ . There $2$ , $3$ , $5$ , $2\cdot2$ , $2\cdot3$ , $2\cdot5$ , $3\cdot5$ , $2\cdot2\cdot3$ , $2\...
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2answers
35 views

can anybody help in finding number of ways the letters of the word 'PERMUTATION' be arranged so that consonants are in alphabetical order? [closed]

I had tried the question and got the answer 11!/(6!2!) but the answer given is 11!/6! if any body can explain that why 2! is not in the answer or the answer is wrong.
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0answers
33 views

What happens to the Permutation Rule when r=0?

This is small but quirky idea that popped into my head in the middle of the night last night. If I have $n$ objects, and want to find out how many permutations (sequences) of $r$ objects there are, ...
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2answers
42 views

Unable to derive reason/formula for permutation problem

What is the probability of $n$ preceding $1$ and $n$ preceding $2$ when we randomly select a permutation of ${1, 2, . . . , n}$ where $n ≥ 4$? I wrote out examples of n! when n equals some number ...
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2answers
84 views

German combinatoric terms vs English terms

I'm a German Computer Science student and I currently work with combinatorics as part of my curriculum. I wanted to research combinatorics in English but I'm confused about the terminology. In German ...
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2answers
42 views

increasing, decreasing, non-decreasing, non-increasing permutation/combination

I have this question and I'm stuck Q: In the set of three-digit integers {100,101,...,999}, how many integers are there (a) with three distinct digits that are either increasing (as in 257, 139) or ...
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2answers
38 views

show that A(T) is a group under operation of composition of functions

Problem: Let T be a nonempty set and A(T) the set of all permuations of T. Show that A(T) is a group under the operation of composition of functions. Permutation of the set T is the bijective ...
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1answer
61 views

There are $5$ apples $10$ mangoes and $15$ oranges in a basket.

There are $5$ apples $10$ mangoes and $15$ oranges in a basket. Then find number of ways of distributing $15$ fruits each to $2$ persons. Can I approach this question as number of ways $15$ fruits ...
2
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2answers
90 views

How many combination of $3$ integers reach given number?

I have 3 numbers $M=10$ $N=5$ $I=2$ Suppose I have been given number $R$ as input that is equal to $40$ so in how many ways these $3$ numbers arrange them selves to reach $40$ e.g. $$10+10+10+...
2
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0answers
20 views

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$ [duplicate]

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$. I'm not quite sure how to start this. The solution starts ...
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1answer
39 views

number of possible subsets formed with odd count of odd numbers = $2^{n_{odd}-1} \cdot 2^{ n_{even}}$

Let $n_{odd}$ represents the number of odd numbers in the set $S$ and $n_{even}$ denote the number of even numbers. The total number of possible subsets formed with odd count of odd numbers $=2^{n_{...
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2answers
80 views

Prove that there is no permutation $\sigma$ such that $\sigma(123)\sigma^{-1}=(124)(567)$

Prove that there is no permutation $\sigma$ such that $\sigma(123)\sigma^{-1}=(124)(567)$. Cycle $(123),(124),$ and $(567)$ has order $3$ so if the equation $\sigma(123)\sigma^{-1}=(124)(567)$ is ...
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2answers
30 views

Number of ways so that at least one soldier find that soldier next to him is also selected.

20 soldiers are standing in a row and their captain want to send 7 out of them for a mission. In how many ways can captain select them such that at least one soldier find that soldier next to him is ...
4
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2answers
73 views

$15$ men and $15$ women into $15$ couples

Find the number of ways of dividing $15$ men and $15$ women into $15$ couples. My solution is: First Man can be paired with any one of $15$ women in $15$ ways. Second Man can be paired in $14$ ways ...