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

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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 ...
-2
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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?
0
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0answers
19 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, ...
5
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3answers
65 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 ...
1
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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 ...
0
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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 ...
0
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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
58 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 ...
0
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0answers
36 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 ...
0
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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 $\textbf{r}_J$ ...
1
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1answer
35 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 ...
2
votes
1answer
60 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
votes
1answer
94 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 ...
0
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1answer
29 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 ...
9
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2answers
153 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 ...
2
votes
1answer
91 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
votes
2answers
40 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 ...
1
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1answer
24 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
44 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 ...
0
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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 ...
0
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3answers
86 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 ...
0
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3answers
57 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? ...
0
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1answer
44 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 ...
1
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1answer
156 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
26 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, ...
2
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2answers
42 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 ...
1
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0answers
37 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 ...
0
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3answers
67 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
77 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.
0
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1answer
19 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 ...
0
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4answers
48 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 $2$ , $3$ , $5$ , $2\cdot2$ , $2\cdot3$ , $2\cdot5$ , $3\cdot5$ , $2\cdot2\cdot3$ , ...
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2answers
34 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.
0
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0answers
28 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, ...
3
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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 ...
2
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2answers
83 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
34 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 ...
0
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2answers
36 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 ...
4
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1answer
58 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
84 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. ...
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 ...
1
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1answer
32 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 ...
1
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2answers
76 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 ...
0
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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
votes
2answers
72 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 $14$ ...
1
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2answers
96 views

How many ways are there to assign $20$ different people to $3$ different rooms with at least $1$ person in each room?

How many ways are there to assign $20$ different people to $3$ different rooms with at least $1$ person in each room? I know how to approach this problem using combinations:$$17!\cdot {3 \choose ...
0
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0answers
9 views

presentation of the symmetric group via transpositions fixing one element

Consider the symmetric group $S_n$. If we use the most popular set of generators $\sigma_1, \sigma_2,\cdots,\sigma_{n-1}$ with $sigma_i$ being the transposition $(i \, i+1)$, it is well known that ...
1
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0answers
28 views

Divison in factoradic base

I'm trying to find am means of dividing two numbers in factoradic base. So far goolging seems to turn up nothing at all. Is there a better way of doing this than long-division? I'm hoping for ...
2
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1answer
35 views

system of two eqautions in three unknowns: finding the number of solutions

I have a system of two equations with three unknowns. $$x+y+z=7$$ $$x+2y+3z=10$$ On solving, I got the following values. $$ y = a$$ $$x = (11-a)/2$$ $$z = (3-a)/2$$ How would I go about finding ...
1
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1answer
37 views

Finding number of possible hands in 5 card stud when order is matter

I'm trying to determine the game 5 card poker when order is matter all my trails ended with fail except the one pair. The way I did one pair assuming it is $\{k,k,x,y,z\}$ $^{13}P_1 \times ^4P_2 ...