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

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Classifying 1 cycle permutation matrices

Given a permutation matrix that is not full rank, is there an algebraic criterion to tell if matrix contains more than one disjoint non-trivial cycle or exactly one non-trivial cycle? Example: ...
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1answer
32 views

Cycle structures of $S_6$

A problem from my algebra homework requests the following: List all the possible cycle structures in $S_6$. For each cycle structure, compute the order of an element with that cycle structure. ...
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2answers
86 views

Prove that any element in $S_n$ can be written as a finite product of the following permutations

Prove that any element in $S_n$ can be written as a finite product of the following permutations: $(a)\ (12),(13), . . . , (1n)$ $(b)\ (12),(23),...,(n−1,n)$ $(c)\ (12),(12\dots n)$. I have no ...
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2answers
61 views

Is $A_n$ non-abelian for $n= 3$?

In the book, it is asked to show that $A_n$ is non-abelian for $n ≥ 4$. Which may imply that it is abelian for $n=3$. Is that so? because $(13)(12)\ne (12)(13)$. Hence is it true to write: $A_n$ is ...
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0answers
31 views

Every odd permutation in Sn can be written as a product of 2n+3 transpositions?

My question is If Set of permutation set Sn for n>3 Prove Every odd permutation in Sn can be written as a product of 2n+3 transpositions and every even permutation as a product of 2n+8 ...
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18 views

Compress Unit ID Permutations

For my multiplayer RTS game, I need to send over the ID's of a player's units selected in a single integer. For every player, there are 16 units, each with an integer ID between 1-16 (inclusive), and ...
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2answers
58 views

Count good sets with given conditions

We need to find the numbers of good sets in a sequence. The good set is: A good set is a sequence of $P$ positive integers which satisfies the following 2 condition : If an integer $L$ appears in a ...
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0answers
45 views

Permutation and Combination- Rowing a Boat

The crew of an 8 member rowing team is to be chosen from 12 men, of which 3 must row on one side only and 2 must row on the other side only. Find the number of ways of arranging the crew with 4 ...
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2answers
166 views

Number of numbers

Given a set of digits {D1, D2, D3 ... DM}, how many sequences of length 'N' are possible with the constraint that a digit 'K' can only appear in the sequence if all the digits less than 'K' present in ...
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4answers
54 views

Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number

Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the ...
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1answer
38 views

Number of elements in cartesian power with a maximum constraint

Problem: I would like to know the number of elements in the cartesian power $X^n$ (cartesian product of one set $X$ by itself, $n$ times) with a maximum constraint: how many elements in $X^n$ have ...
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2answers
137 views

Number of elements in cartesian power with a majority constraint

Problem: I would like to know the number of elements in the cartesian power $X^n$ (cartesian product of one set $X$ by itself, $n$ times) with a majority constraint: how many elements in $X^n$ have a ...
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0answers
38 views

How many $4$ digit numbers with distinct digits can be formed using $0,1,2,3,4,5$?

So left most digit can be filled with $5$ (can't use $0$ there) then next one got $5$ option then $4$ and $3$. So the answer is $5\cdot 5\cdot 4\cdot 3 =300$. Is it correct ?
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1answer
11 views

Combinations - odd num with different start and end

I am having problems with the following combinations problem: Given a number $1243356$, how many ways are there to form an odd number with different start and end digits? My approach is to split the ...
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0answers
27 views

Number of Words

In an alphabet there are 3 consonants and 5 wovels. 1 letter words are meaningless. The words that have two consonants together are meaningless. The words that have 3 vowels together are meaningless. ...
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2answers
95 views

Password Permutation/Combination Problem

So here's the problem: At a certain company, passwords must be from 3-5 symbols long and composed of the 26 letters of the alphabet, the ten digits 0-9, and the 14 symbols ...
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1answer
22 views

Cyclic permutation of conditional probabilities

Consider a generic, strictly positive, joint probability distribution $p(x,y,z)$. Is the following formula: $$ p(x|y)\,p(y|z)\,p(z|x) = p(y|x)\,p(z|y)\,p(x|z) $$ always true? I can prove it in ...
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1answer
44 views

Find number of ways to choose 3 vertices

Consider a $2n+1$ sided regular polygon.In how many ways can we choose $3$ vertices out of these $2n+1$ vertices so that the centre of the polygon always lies inside the triangle formed by joining ...
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2answers
30 views

Solve for x, permutation multiplication

Could anyone help me out with this? I'm supposed to solve the equation for $x$. $$(1 3 4)x(32)=(1 2 3 4) \textrm{ in } S_4$$
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1answer
40 views

filling up numbers in a matrix

Suppose you have a $k.n \times 2$ matrix. You have to fill up the numbers $1,2,3, \cdots, n$ as entries in such a way that in each column it is non-decreasing, in each row it is strictly increasing ...
3
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1answer
77 views

Isomorphic but not equivalent actions of a group G

This is in some sense a continuation of this problem. Given a group $G$ I would like to exhibit two actions of $G$ on a set $[n] =\{1,\ldots,n\}$ such that the two actions are isomorphic yet not ...
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1answer
64 views

Who is honest? Who lies? [duplicate]

A bioinformatics department at a University has 100 professors- some are honest and hardworking, while others are deceitful & do not like students. The honest prof. always tell the truth, while ...
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0answers
30 views

Find value of K?

IF the number of ways of selecting K coupons out of an unlimited number of coupons bearing the letters A, T, M so that they cannot be used to spell to the word MAT is 93, then what is the value of k.
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1answer
63 views

If we are given that a list of $n$ numbers has $11,660$ derangements, what is the value of $n$?

The Full Question For the positive integers $1,2,3,\dots n-1,n$, there are $11,660$ where $1,2,3,4,5$ appear in the first five positions. What is the value of $n$? My Work First I considered all ...
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0answers
43 views

Ways of Assigning Grades to Students

I was reading somewhere that the number of ways 10 students could be assigned the grades A,B,C,F would be $4^{10}$. However, that seems to be counting sequences such as AABCFBFAFC and switching the ...
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2answers
29 views

How many ways to select 5 cards with at least one king.

I am given a deck of $52$ cards in which I have to select $5$ card which has at least one king. So I selected one king out of $4$ and then remaining from deck as $$^4C_1. ^{51}C_4$$ which however ...
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1answer
27 views

4 People Gift Exchange

4 people are exchanging gifts. How many combinations are there so that no one receives their own gift? I tried this problem myself, and got 3!. My friends told me that it's 9. I got 3! because I ...
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1answer
42 views

Probability of winning a certain states lottery.

A certain state's lottery consists of choosing $6$ numbers at random without replacement from the set $\{1,2,3,\ldots,40\}$. What is the probability that someone wins the jackpot? I just want to ...
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3answers
174 views

Sigma of Permutations

Given a permutation p of numbers 1, 2, ..., n. Let's define $f(p)$ as the following sum: $$\large f(p)=\sum_{i=1}^n\sum_{j=i}^n\min({\rm p}_i,{\rm p}_{i+1},...,{\rm p}_j)$$ What is the exact job of ...
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1answer
75 views

Why 312 Avoiding?

I have recently had the chance to attend a nice talk in Combinatorics, and once the speaker alluded to the famous 312-avoiding pattern problem, I was reminded of the following question I have had ...
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1answer
49 views

How to count the ways to induce a permutation?

I'm reading a recreational book about combinatorics, that discusses, in passing, the ways to 'induce' a permutation of the index set {1,2,3,4}. The book notes that there is exactly: 1 way to ...
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1answer
70 views

Count permutations with given cost and divisbilty

I am given $N$ . We need to count such permutations of $N$ numbers with each element between $0$ to $9$ which satisfy following conditions : Cost of permutation is less than or equal to given $M$. ...
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2answers
62 views

Find the order of $\sigma^{1000}$ where $\sigma$ is the permutation (1,3,8)(2,7)(4,9,6,5)

My book doesn't have any examples of how to do this, so I'm a little lost. I know the order of a permutation is the lowest common multiple of the lengths of its disjoint factors, and so the order of ...
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0answers
10 views

Trascendental numbers permutation

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number's sequence of decimal digits through a ...
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3answers
24 views

How many way can 3 distinct letter and 2 distinct digit be arranged if the digit must be together

Hi mind helping me out for this question ? How many way can 3 distinct letter and 2 distinct digit be arranged if the digit must be together. Thanks.
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1answer
86 views

The normaliser of the left regular image [D&F]

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 186): Let $H$ be a group of order $n$, let $K=\text{Aut}(H)$ and form $G=\text{Hol}(H)=H \rtimes K$ (where ...
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Problem about combinations and permutations

A rent-a-car company has 14 available cars on its lot and 3 customers in the office. In how many ways can the customers be assigned to cars? The answer i get is 364 from 14c3 but its wrong.. Any ...
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1answer
31 views

Question about Permutation Sets (Groups and Symmetries)

Let $a = (123)(456)$ in $S_{10}$. Find the highest possible order of a permutation $b$ in $S_{10}$ such that $b^k=a$ for some $k$. Attempt: I already know that there are only two possible cases for ...
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2answers
108 views

Probability that six girls sit together?

Six boys and six girls sit in a row randomly. Find the probability that six girls sit together? (a)$\frac{1}{32} (b)\frac{2}{7}$ (c)$\frac{5}{12}$ (d) None of these what i have tried Since six ...
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0answers
17 views

What is the maximum possible identicons count on GitHub?

GitHub provides default identicons as the profile picture. You can get one for your account too, like this. There are many other implementations on GitHub, using the similar way and producing similar ...
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22 views

Composing multiple permutations

Let $R$ and $F$ be $2$ permutations. If I want to calculate $R^3*F$ what is the right order? I know I should compose them from right to left but should I first calculate $R^3$ and then compose it ...
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2answers
45 views

A and B likely to contradict each other in stating same fact?

A speaks truth in $75\%$ cases and B in $80\%$ cases. In what percentage of cases are they likely to contradict each other in stating the same fact? (a) $70\%$ (b)$35\%$ (c) $25\%$ (d)$20\%$ what i ...
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1answer
44 views

Permutation isomorphic subgroups of $S_n$ are conjugate

Consider $G,H \leq S_n$ and their natural action on $[n] = \{1,\ldots,n\}.$ We say that $G$ and $H$ are permutation isomorphic if there is a bijection $\varphi:[n] \mapsto [n]$ and group isomorphism ...
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0answers
37 views

Number of integer partitions of a natural number [duplicate]

Consider $S \in \mathbb{N}$ where $\mathbb{N}$ is the set of natural numbers. I want to obtain $S$ by summing $k$ natural numbers strictly positive. I want to find how many possibilities I have to ...
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1answer
21 views

Distribution of different objects among person [closed]

Eight different balls are to be distributed among 6 boys such that each boy gets at least one. How many ways are there to distribute the balls?
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69 views

Can $S_4$ be generated by $(1,2),(1,3),(1,4)$?

I am having troubles answering it. It sometimes confuses me. I could really use some help.
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2answers
66 views

Probability that one will be mathematician and the other physicist?

From a group of $13$ scientists which contain $5$ mathematicians and $8$ physicists, it is required to appoint a committee of two. If the selection is made without knowing the identity of the ...
3
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1answer
69 views

Permutation and combination with overcounting

A lady gives a dinner party to six quests. The number of ways in which they may be selected from among ten friends, if two of the friends will not attend the party together is (a) $112$ (b) $140$ ...
3
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2answers
28 views

Finding $\gamma \in S_7$ that satisfies $\gamma^4=(3412675)$

I have proved it using an arbitrary $\tau=(1234567)$, taking it to the fourth power, and finding that $\tau^4=(1526374)$. Can I just see the pattern of where each element went and match it to the ...
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0answers
35 views

Multinomial problem

Suppose one has a nested table of disitnct primes then the permutations of their products produce dupluicates at certain values. For example, letting the primes $\{a,b\}=\{2, 3\}$, the products of ...