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

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Finding the initial permutation

Assume a permutation P[1], P[2], ..., P[n]. Now we have made N permutations Q[1], Q[2], ..., Q[N], Q[i] is permutation P without element i (we subtract 1 from all elements bigger than i). For ...
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1answer
35 views

Arrangements of the word HULLABALOO

Three problems below with my attempt solutions: 1) How many ways (ordered selections) can the letters of the word HULLABALOO be arranged? $$\frac{10!}{3!2!2!}$$ 2) How many distinguishable ...
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1answer
27 views

Shuffling cards alternately return to original position

There are $2n$ cards in a stack. At any stage, if the cards are $a_1,a_2,\ldots,a_{2n}$ from top to bottom, then it becomes $a_{2n},a_1,a_{2n-1},a_2,\ldots,a_{n+1},a_n$. In how many steps will the ...
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1answer
33 views

Counting the number of permutations where $i$ does not follow $i-1$

Whilst reviewing, I've found a problem where the book has an answer, but no suitable explanation, and I can't begin to connect the two. The problem is simply as follows: Count the number of ...
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1answer
24 views

Product of differences of permutation in circle

Numbers $1,2,\ldots,n$ are arranged into a circle. What is the maximum product of the differences $|x_1-x_2|\times|x_2-x_3|\times\cdots\times|x_{n-1}-x_n|\times|x_n-x_1|$? I think the maximum should ...
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2answers
39 views

a question about abstract algebra, a question related to permutation

Given $X=\{1,2,......n\}$, let us call a permutation $p$ of $X$ an adjacency if it is a transposition of the form $(i,i+1)$ for $i\le n-1$. Prove that $(i,j)$ is a product of an odd number of ...
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1answer
48 views

Sum of differences of permutation in circle

Numbers $1,2,\ldots,n$ are arranged into a circle. What is the maximum sum of the differences $|x_1-x_2|+|x_2-x_3|+\ldots+|x_{n-1}-x_n|+|x_n-x_1|$? I think the maximum should occur when the numbers ...
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2answers
44 views

Do these permutations commute?

I read in my textbook that disjoint cycles commute. But what about $(1,2,3) = (1,3)(1,2)$ is it equal to $(1,2)(1,3) = (1,3,2)$ although they are not disjoint? And, if they are not equal how come in ...
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3answers
62 views

Recovering the original values from given information.

We have some N numbers[1..N] and N students. Originally we assign each number to single student. Call this assignment as the initial state of the assignments. Instance of assignment is described as ...
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1answer
63 views

In $S_n$, if $ε = α_1α_2 \cdots α_r$ where $α_i$ is a $2$-cycle, then $r$ is even.

In $S_n$, if $ε = α_1α_2 \cdots α_r$ where $α_i$ is a $2$-cycle, then $r$ is even. I don't know how to start. Note, $ε$ is the identity of the permutation group $S_n$.
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37 views

Identifying this kind of combinatoric permutation

I feel kind of silly asking this, but I am having a hard time identifying what this exactly called. I'm specifically trying to find the wiki page on it and could not find it on this list of ...
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2answers
45 views

Count how many numbers containing number 4 among the numbers 1 to 8900?

I Got a Problem like this "Count how many numbers containing number 4 among the numbers 1 to 8900 ?" what is the appropriate formula to solve the problem above, I tried to solve it by creating a ...
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0answers
33 views

The number of Permutation polynomial in a field

I need to know,please: (1) How many permutation polynomial exist in a finite field (any field)? (2) Is there any way to pick a random permutation polynomial in this field?
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1answer
17 views

How many different combinations are possible

Say there are 4 stations for ice cream toppings. station1: 3 choices station2: 2 choices station3: 2 choices station4: 3 choices How many different combinations ...
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37 views

Prove that for each $\sigma \in Aut(S_n)$ $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), …, \sigma(1,n)=(a,b_n)$ [duplicate]

Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), ...., \sigma(1,n)=(a,b_n)$ for some distinct integers $a,b_1,....b_n \in \{1,...,n\}$. I was trying ...
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2answers
22 views

count number of permutation that map no even number to itself

Problem: How many permutations of numbers $1, 2, ..., 10$ exist that map no even number to itself? I understand that this is "Hatcheck lady" problem. But I am a bit confused how to solve it. So my ...
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0answers
22 views

How to to minimize a sum by changing summation order

I have two vectors $(x_1,\dots,x_n),(y_1,\dots,y_n) \in \mathbb{R}^{n}$. I want to find a permutation $\sigma$ such that $$ \sum_{i=1}^n |x_i -y_{\sigma(i)}|^2$$ is minimized. Is there a better way ...
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1answer
55 views

In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. …

Problem : In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. Find the number of ways in which the candidate can score 60 % marks ...
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1answer
36 views

Binomial Coefficient Combinations

I have tried to figure this out and I cannot. The professor gave us an answer of 13,536 but I do not see any way in which he got to his answer. Any help would be greatly appreciated. A certain ...
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1answer
38 views

If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the …

Problem : If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the number of ordered pairs (p,q)? Can we use this : let $r^2s^4t^2$ = ...
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1answer
47 views

Exponential Generating Function for Permutation with no Fixed Points

While reviewing, I've come across a problem that seems to outline my lack of knowledge with regards to (specifically exponential) generating functions. For some reason, I understand "ordinary" ...
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1answer
48 views

Permutation Matrix Proof

This is off a study guide for an exam I have in about two days. I really don't understand the problem entirely and would appreciate and help. Let $\sigma \in S_{n}$, where $\sigma$ is a permutation ...
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2answers
26 views

Four letters are picked from the word BREAKDOWN. What is the probability that there is at least one vowel among the letters?

I know that the right answer is 0.881 but i'm not sure how to get there.
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34 views

Find the number of ways in which 10 different books can be shared between a boy and a girl if each is to receive an even number of books.

The right answer is 510 but my calculations keep giving me 252. where did I go wrong? please be thorough because I really do not understand this topic.
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1answer
113 views

Generalization of a property Of $A_5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_5$ and $\langle H,K\rangle < A_5$. We can show that there exists a maximal subgroup $M$ of $A_5$ such that ...
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2answers
33 views

Combinatorics For $4$ Pool Balls

There lie $4$ pool balls on a pool table: two striped and two plain. Two of the pool balls are selected at the same time, at random. Given that one of the selected balls is striped, what's the ...
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1answer
63 views

Words of the Normal Form of the Presentation of a Finite Monoid

Massive Edit: After consulting with a few mathematicians at my university, I got a better understanding of what I was actually looking for. $$ \langle\ s,\ t\ \vert\ s^2 = 1,\ t^n = 1\ \rangle $$ ...
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1answer
73 views

How many ternary strings of length 4 have exactly one 1?

Answer: Ternary strings have symbols 0, 1, and 2. If there is exactly one 1, then there are 3 positions the one can be in and 2*2*2 ways to fill the other 3 blanks with a 0 or a 2. So the answer is ...
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33 views

Circular array with minimum absolute difference among adjacent elements

Given a circular array, rearrange the array so that the maximum absolute difference between adjacent elements among all elements is minimum. Can anyone help me with this?
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32 views

“Stars and Bars” method when all variables are multiple of same number

Find the number of non negative integral solutions of the equation $x + y + z + u = 100$, where $x, y, z$ and $u$ are multiple of $5.$ My approach using stars and bars: Since all are multiples of ...
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1answer
51 views

A Proof Question

Prove : $$\sum_{k=1}^nkp(n,k)=n!\;,$$ where $p(n,k)$ is the number of permutations of $\{1,2,\ldots, n\}$ which have exactly $k$ fixed points. I was using $$p(n,k) = \frac{n!}{(n-p)!}$$ and ...
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4answers
155 views

Problem on selecting group of card from a well shuffled pack of card

I have a problem I'm working on: The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 card to guarantee that three cards are from some same suit is which amount? I got ...
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1answer
44 views

Show that the permutation [n, n-1,…, 2,1] has n(n-1) inversions

Show that the permutation $[n, n-1,..., 2,1]$ has $n(n-1)$ inversions How do I show that this is true? Why isn't $(n(n-1))/2$
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2answers
67 views

Find the number of 3 letter words that can be formed from the word 'SERIES'.

To find the number of three letter words that can be formed from the word 'SERIES', with or without meaning and without repetition. The number of permutations if all letters were distinct = ...
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1answer
38 views

Permutation of $n$ women and $m$ men, in a line, where the women dont get along with each other

So the $n$ women can't sit next to each other. So in a straight line how many ways can they be seated? I know this problem is partitioning distinct balls in $n+1$ partitions, out of which $n-1$ of ...
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737 views

Even Number cards?

There are $15$ cards on a table, marked with an integer $1$ from to $15$ . How many ways can I take cards such that the sum of the numbers on the cards is even? Please help me?
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34 views

how to calculate these intersections without having to count all combinations

We have the following sets: $X= {(a,b,c,d) ∈S: b< c < d},$ $Y= {(a,b,c,d) ∈S: a< c < d},$ $Z= {(a,b,c,d) ∈S: a< b < d},$ $F= {(a,b,c,d) ∈S: a< b < c},$ Where each of ...
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15 views

How can I prove that every group of order 4 is Abelian. [duplicate]

Prove that every group of order 4 is Abelian. I heard the proof is just 3 lines but I don't know how to proceed. I tried proving it by showing it is isomorphic to a group of permutations, but got ...
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1answer
281 views

Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
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18 views

Amount of inversions in permutations.

Let $I_n(k)$ be the number of permutations of $n$ values that have exactly $k$ inversions. The true is expression: $$I_n(k) = I_n\left( \binom n 2 - k\right) $$ but I don't understand why. Please help ...
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523 views

Math puzzle: 10 digit strings generations

There was a question in a math competition that I attended last year. At the end of competition, I realized that my answer was wrong for the question below and I have never been able to figure out how ...
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34 views

Find a permutation with the given square or cube

Problem: find a permutation such that $x^2 = (1\;3\;4\;5\;7)$, $x\in S_7$ $x^3 = (1\;3\;4\;5\;7)$, $x\in S_7$ Must find all possible solutions for $x$. Progress I have solved for the first ...
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1answer
39 views

Number of ways of choosing identical balls

Suppose we are given a bag of $n$ identical red balls, what is the number of ways of choosing $3$ red balls from the bag? I know the answer is $$ \binom n3 $$ but isn't there just one way of choosing ...
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1answer
35 views

Number of $k$-cycles in $S_n$

I've computed that the number of $k$-cycles in $S_n$ is $\frac{n!}{(n-k)!k}$ and wiki seems to agree with me. Now, we know that in $S_n$ the number of $k$-cycles is also equal to the cardinality of ...
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81 views

A question on colouring cubes

We are given 6 distinct colours and a cube.We have to colour each face with one of the six colours and two faces with a common edge must be coloured with different colours.How many distinct ...
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2answers
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Permutations/Integer Solutions to Equations

I'm pretty lost on this so I'd appreciate some feedback as to whether or not I'm on the right track. Find the number of integer solutions of $x_1 + x_2 + x_3 = 15$ subject to the conditions $0 \le ...
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Permutation and symmetric Group

I have the following task: For all $i,j \in \{1,2,3,4\} $ such that $i+j=5$, let $G$ be the set of permutations $ \sigma \in S_4 $ satisfying $\sigma (i) + \sigma(j)=5$. a.) List all the ...
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2answers
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How can it be proven that a cycle of length k is an even permutation if and only if k is odd?

How can it be proven that a cycle of length k is an even permutation if and only if k is odd? I know it can be done using the fact that a permutation which exchanges two elements but leaves the rest ...
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1answer
26 views

question relate to repetition, permutation [closed]

1) a) show that $2^n = \sum_{r=0}^n {n \choose r}$ b) in how many ways can 8 boys be devided into two unequal sets? my solution: 8P2 =56 2) to win a prize in a 'scratchit' game te ticket must show ...
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20 views

repetition of permutations problems bingo

for a game bingo, organizers place one marble with 0 marble on it, one marble with 1, and one marble with 2 and so on up to one marble with 9. Each time a number is called one number is drawn, the ...