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

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2
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2answers
79 views

How do I calculate permutations where some values are restricted?

I am curious about the formula for determining the number of combinations there are in a given set where some values are restricted to a certain range. For example, if I have a 10 character, ...
0
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0answers
74 views

Summing the product of combinations of matrix elements

I have a situation where I have an $NxN$ matrix $A$ where each element $a_{i,j}\in\mathbb{R}_{\leq 0}$. I would like to consider the set of all collections of elements such that each collection of $N$ ...
1
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2answers
635 views

Give the digits $0, 1, 2, 3, 4$, and $5$. How many four digit numbers can be formed if digits can be repeated and contain at least one digit $3$

Given the digits $0, 1, 2, 3, 4$, and $5$. How many four digit numbers can be formed if digits can be repeated and contain at least one digit $3$?
3
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1answer
398 views

Distributing $n$ different things among $r$ persons

How can $10$ different pencils be distributed among $3$ students? MY TRY $1$ total ways $= 3^{10}$ MY TRY $2$ $10 \times 9 \times 8 =720$ Which one is correct? If both are wrong what is correct ...
0
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2answers
69 views

Permutation Homework

There are two teams.Two games were played.There are three possible outcomes which are win, lose or draw. how many permutations are there?
2
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1answer
30 views

Permutation query

Would anyone be able to help here with this one ? Let $A = \{a, …, z, A, …, Z, 0, …, 9\}$ be some alphabet and let $$q = q_1, …, q_m \text{ and } w = w_1, …, w_n$$ be finite-length words in $A^*$. ...
1
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1answer
31 views

Should I divide this permutation problem into cases or are there any quicker methods?

I have got an idea for the second question but I think my approach is too long and I would like to ask whether there are any other quicker methods? Eight cards are selected with replacement from a ...
0
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2answers
2k views

Outcome possibilities with three teams and three outcomes for each game

So there are six teams (let's say: 1,2,3,4,5,6), and they pair up to face each other, (so three games in total). In each game, one team either wins or their is a tie. Let's set up the teams and their ...
3
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2answers
92 views

Find permutation that solves $\;\tau \circ X = \sigma$

I need to find a permutation $X$ that solves $\;\tau \circ X = \sigma,\;$ given $$\tau = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 5 & 2 & 1 \end{bmatrix} = ...
0
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2answers
36 views

Partioning/Enumeration

How many ways can one distribute A) 15 Balls into 3 bags. Both bag and balls are distinct (labelled) and each bag must contain at least one ball. B) 10 balls into 3 bags. again both bag and balls ...
0
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1answer
38 views

Number of k-permutations that have odd number of an element

I want to find a recurrence relation $h_k$ for the number of k-permutations of $\{\infty a,\infty b, \infty c, \infty d \}$ that have an odd number of a's. I let $h_0=0$ because there is no odd ...
0
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3answers
989 views

Combination of Password Question

Suppose that a password for a computer system must have at least 8, but not more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a ...
0
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1answer
73 views

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 ...
0
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2answers
126 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 ...
1
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1answer
70 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 ...
0
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1answer
65 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 ...
1
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2answers
51 views

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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2answers
70 views

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 ...
2
votes
1answer
105 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 ...
2
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0answers
143 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 ...
5
votes
2answers
215 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
0
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1answer
153 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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2answers
400 views

ln how manyways can we distribute $7$ apples and $6$ oranges among $4$ children so that each child gets at least 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
96 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 ...
0
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2answers
363 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 ...
0
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1answer
44 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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2answers
70 views

Abstract Algebra - Permutations

I'm asked to show that $(1,2,3) \in S_3$ generates a subgroup which is normal. I know that I could show it explicitly but that would be tedious. I think it may have to do with the fact that $(1,2,3)$ ...
2
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0answers
83 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 ...
4
votes
3answers
539 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
24 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
512 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 ...
0
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1answer
93 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 ...
0
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2answers
81 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 ...
1
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1answer
68 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 ...
0
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1answer
233 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 ...
0
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1answer
87 views

Let $\sigma\in S_n$. Show that $\sigma(1\:2\:\dots\:n)\sigma^{-1}=(\sigma(1)\:\sigma(2)\:\dots\:\sigma(n))$.

I have no idea how to show this as cycle notation really confuses me.
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2answers
3k 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 ...
-3
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2answers
249 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
154 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
5k views

Permutation & Combination card sequence . .

I've been trying to do these 2 questions about Permutation & Combination which linked to card play. Q1 says : ...
3
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2answers
923 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
67 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 ...
1
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1answer
51 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 ...
0
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2answers
113 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
126 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
91 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. ...
3
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0answers
124 views

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 ...
0
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3answers
145 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.
2
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2answers
324 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.
2
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1answer
421 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 ...