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

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Alternating Pair

I want to find the number of permutations of $1,2,\ldots,N$ having exactly $k$ triples satisfying the condition that either $n_{i-1}>n_i<n_{i+1}$ or $n_{i-1}<n_i>n_{i+1}.$ For example for ...
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2answers
32 views

List of all elements of $A_4$ - Jamie Mulholland p. 85

p. 72: $m$-cycle $\iff m - 1$ transpositions. Hence 3-cycle $\iff 2$ transpositions. I condone all the calculations overhead, but I don't understand the proof blueprint. (1.) How do you ...
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1answer
54 views

Finding Required Permutation

I have numbers from $1$..$n$. I want to find number of permutation from all $n!$ permutation where the numbers have following arrangement. $L$ $G$ $L$ $G$ $L$ or $G$ $L$ $G$ $L$ $G$. Where L means ...
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1answer
35 views

Total number of ways to color a regular graph.

I have problem stating "Find total number of ways to color a regular pentagon with 5 colors." If we consider(Exact 5 colors to color the graph) it unlabeled graph then it will be the same to ...
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2answers
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Fraleigh Section 9 Question 27

Question 27 on Section 9 of Fraleigh 7th edition: Part (a) Asks us to prove that a permutation in $S_n$ can be written as a product of at most $n - 1$ transpositions. I feel that this is not true. ...
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Permutation/Combination Question

A three digit number is to be formed by using the digit from 1 to 9 without repetition, find the number of three digit numbers that can be formed if the units digit is an odd number, the hundreds ...
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1answer
87 views

Expected Value of this function

Let’s consider a random permutation p1, p2, …, pN of numbers 1, 2, …, N and Function F is calculated as F=(X[2]+…+X[N-1])^K, where ...
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0answers
58 views

Number of permutations when combining two sets?

I have two sets $\{a_{1},\ldots,a_{K}\}$ and $\{b_{1},\ldots,b_{L}\}$, where I know that $a_{1} \leq a_{2} \leq \cdots \leq a_{K}$ and $b_{1} \leq b_{2} \leq \cdots \leq b_{L}$, but do not know the ...
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1answer
35 views

Number of permutations for n elements with different probabilities

I'm studying the paper Database-friendly random projections: Johnson-Lindenstrauss with binary coins by D. Achlioptas and can't manage to work out the total number of permutations with repetitions in ...
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1answer
70 views

ln how manyways can we distribute 7 apples and 6 oranges among 4 children so that each child gets atleast one apple.

In how many ways can we distribute 7 apples and 6 oranges among 4 children so that each child gets at least one apple? I think this can be solved by using permutations because the word ...
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0answers
33 views

A company has 20 employees, 12 male and 8 female. How many ways are there to form a 5 person committee?

A company has 20 employees, 12 male and 8 female. How many ways are there to form a 5 person committee that contains at least one male and at least one female? Is this right? no. of ways to select 5 ...
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2answers
93 views

In how many ways can we arrange 40 boys and 20 girls in 5 groups of 12 members each, so that each group contains at least one girl.

My approach There are 5 groups with 12 members each,so if there was condition like there should be 3 girls and 2 boys i would do (20C3)*(40C2) But here it is given as atleast one girl,how to ...
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1answer
41 views

K×H is Isomorphic to A4?

Prove \ Disapprove : There exist two non-trivial sub groups $K$ and $H$ such that $A_4\cong K×H$: My intuition was to disprove this claim by saying that $H$ or $K$ must be the Klein sub-group and ...
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0answers
38 views

Removing the Summation (Closed Form)

The following question from "Combinatorics of Permutations" : $$ E[X] = \sum\limits_{k = 2}^n \frac{k\cdot T(n,k)}{n!} $$ where $$ T(n,k) = k \cdot T(n-1, k) + 2 \cdot T(n-1, k-1) + (n-k) \cdot ...
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0answers
65 views

How many different ways can 10 octupuses touch legs?

There are 10 octopuses (octopi?). Each octopus has 8 legs. Legs on an octopus can only touch touch legs on other octupuses. Assuming each leg touches exactly 1 other leg, how many different ...
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2answers
191 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
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1answer
19 views

Combination - Ordering Boy Scouts

In how many different ways can 9 distinct boy scouts be arranged in a 3 × 3 formation? In such a formation, there are 3 scouts in the first row, 3 in the second, and 3 in the third. Two formations are ...
3
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4answers
167 views

Permutations of a word with repetitions and conditions

How many permutations of "committee" exist where is must not end in an 'e' ? I've been trying to figure out a possible angle of attack on this question. I've tried to say instead, "how many ...
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0answers
34 views

How many permutations of the sequence 1, 2, 3…N where none of the first K numbers in the original sequence is in it's place?

For the sequence 1, 2, 3 ... N there are of-course N! permutations. But for a given K, where 1 < K ≤ N how many permutations are there given none of ...
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2answers
52 views

Prove this is a subgroup: Subset of $S(A)$ consisting of all the permutations $f(a) = a$

Let $A$ be a set and $a \in A$. Let $G$ be the subset of $S(A)$ consisting of all the permutations $f$ of $A$ such that $f(a)=a$. Prove that $G$ is a subgroup of $S(A)$. I really have no clue how to ...
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1answer
33 views

How many permutations have the common minimum in the intersection of $2$ subsets of a set of $1^{st}\ n$ natural numbers

Given a set of elements $N=\{1,2,\ldots,n\}$and two arbitrary subsets $A\subseteq N$ and $B\subseteq N$, how many of the $n!$ permutations $\pi$ from $N\to N$ satisfy $min(\pi(A))=min(\pi(B))$, where ...
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1answer
79 views

In how many ways can the Letters of the Alphabet be permuted such that it does not contain CAR,DOG,PUN,BYTE

Im using the principle on inclusion and exclusion to solve this There are 4 cases C1,C2,C3 ,C4 respectively So taking the case where CAR DOG and BYTE comes together ...
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2answers
164 views

How many 3 digit even numbers are there(No Repetition)?

First find numbers ending with 0 So, 1's place-1 10's place-9 100's place-7 (2 digits are already consumed and 0 can't be used) So ...
0
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1answer
27 views

Counting the number of permutations and combinations

How many numbers consisting of five digits each can be made from the digits $1,2,3,\ldots,9$ if (a) the numbers must be odd (b) the first two digits of each number are even. ...
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2answers
124 views

What is the number of permutations for given N numbers, such that the first part is non-decreasing?

Let $A$ be a list of $n$ numbers in range $[1,100]$ (numbers can repeat). I'm looking for the number of permutations of $A$ which start with a non-decreasing part, where this part ends with the first ...
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1answer
44 views

Permutations using coefficient method [duplicate]

I had a question which is as follows:Number of words of 4 letters formed using the word IITJEE.The book says the answer as coefficient of $x^4$ in 4!$\mathrm{[1+ \frac ...
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2answers
153 views

Permutation & Combination card sequence . .

I've been trying to do these 2 questions about Permutation & Combination which linked to card play. Q1 says : ...
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2answers
124 views

How many ways to split n elements in k groups? [duplicate]

The order of the groups does not matter The size of group must be at least 1 For example, in a more specific question How many ways to split 5 number in 2 groups?, we got the answer 15 from Jared, ...
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2answers
49 views

Ways to select three-man teams

In a competition there are 18 competitors. Answer the following: A) During the first day they're competing in three-man teams (total of 6 teams). How many ways are there to select the teams? B) If ...
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1answer
49 views

A question on a matrix built with permutations of the $n$ first integers.

Let each row and each column of a $n \times n$ matrix $A$ be a permutation of $\{1, 2,...n\}$ and let $A$ be symmetric. (a) If $n$ is odd, prove that each of $1, 2,..., n$ occurs on the principle ...
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2answers
32 views

Perfect shuffle of 52 cards

Prove: How many perfect shuffles of a deck of 52 cards do you need to do until the deck returns to its original order? Can anyone please help me prove this? Attempt: I have tried putting the deck of ...
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1answer
59 views

Explanation of basic definitions in game theory.

In the article entitled Non-Cooperative Game written by Nash in 1951, he discussed about the symmetries of games. Due to my lack of basic knowledge in permutations and symmetries, I looked up some ...
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1answer
42 views

Permuation, disjoint cycles proof by induction.

I am having a hard time writing out a general proof. Can anyone please help? Thank you. Exercise: Show that any k-cycle (a1,......,ak) can be written as a product of some number of (k-1) 2-cycles. ...
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Decomposition of disjoint cycles

Work out the decomposition in disjoint cycles for the following. a) (14)(12345) = (15)(234) b) (12)(2345) = (12345) c) (12)(23)(34) = (14)(24) d) (13)(1234)(13) = (143)(2) Can anyone tell me ...
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3answers
29 views

Permutations 1-line notation, and inverse

Write (15)(286)(479) in 1-line notation. Find the inverse of (15)(286)(479). Can anyone please help? Thank you.
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2answers
86 views

Show that $\sigma^2$ is a Cycle iff the length of $\sigma$ is Odd

I got this question. I'm totally stumped and I don't know what to do. Let $\sigma$ be a cycle of length $k > 2$. Show that $\sigma^2$ is a cycle iff $k$ is odd.
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1answer
45 views

Why is the order of a $k$-cycle $\sigma$ equal to $k$?

If $\sigma = (a_1,a_2,\ldots,a_k)$, then the order of $\sigma$ is $k$ because $\sigma^k = Id$. I've tried finding a proof on the internet, but all sources just say "it's clear that", etc. I'm probably ...
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2answers
124 views

combination and permutation !!!!

I have 3 questions that i had a try to do but i didn't understand them could anybody please help me to solve these questions. For Q1 i know how to use the multiplication counting procedures for a) i ...
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1answer
98 views

Exotic 6-horse race betting probabilities

I'm gearing up for horse racing season, and I'm trying to teach some fellow engineering friends how to bet "exotic" bets by using colored dice to simulate horses. So, the odds for each horse winning ...
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2answers
29 views

Disjoint cycle permutations

Work out the decomposition in disjoint cycles. I am working on disjoint cycles. I sometimes get confuse, so can anyone please check my work. A) $(13)(2345) $ I am starting from right to left So $2$ ...
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1answer
19 views

Permutation problem in language

A language has 28 letters. Each word in the language consist of maximum 7 letters. How many maximum combinations letters can be framed. Provided the letters can be repeated.
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1answer
37 views

Combinations or Permutations of bits

I am a computer science major and was explaining to someone how a computer uses bits to represent numbers. If you have 1 bit, you can have 0 or 1. With 2 bits, you can have 00, 01, 10, 11, or 0, 1, ...
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0answers
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Symmetric Group $S_3$

I just wanted to make sure I am thinking about this correctly. I would like to take the product $(123)(231)$. Here, $2-3-1,3-1-2,1-2-3 \Rightarrow (123)(231)=(132)$.
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Permutations product

Can anyone please help me compute $513642798 \times 971265384$ attempt: I start with the right permutation, So $9 \to 7 \to 9 = (97)$ however, after that do i go to $7 \to 1 \to 3$ ...? I am ...
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4answers
62 views

Permutation on word if E,F,G have to stay in order

Im stuck on a problem which I have answered and need help to verifiy if I have done/understood it correctly. Problem If we have the following string: ...
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1answer
32 views

permutations of “optimization”

How many permutations of the word optimization are there? I get confused with the repetition of the letters. If all 12 letters were distinct, then we would have 12! Because 4 letters are repeated, I ...
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0answers
231 views

Finding the number of arrangement of N people of different height such that K of them are visible from front

Moderator Note: This is a current contest question on codechef.com. [Initially, I had asked this question in stackoverflow, but someone suggested to post it here, and hence this question is ...
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1answer
270 views

Find total perfect combinations

Suppose we are given N numbers and a value K. Now we can interchange the positions of numbers to form different combinations, where if there are two same numbers then their arrangments will be ...
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2answers
39 views

combinatorics - permutations question, possibly with pigeon hole

Let $A \in Mat_n(\mathbb R)$ such that $\forall i,j: a_{ij}\geq 0$ We are given: $$\forall j: \sum_{i=1}^n a_{ij}=\sum_{i=1}^n a_{ji}=1$$ show there's a permutation $\pi \in S_n$ such that $$\forall ...
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1answer
36 views

Permutation of a 4 character string made up of letters and numbers

This is a straightforward question but I didn't pay attention in school. I want to know how many permutations there are for a 4 charcter string made up of numbers and letters. After a quick look ...