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

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Partially ordered permutations

If I have a set $X$ of length five such that $X=\{1,2,3,4,5\}.$ I want to find the number of permutations where the relative order of the $2$ sets $\{2,4\}$ and $\{2,5\}$ is maintained. In other ...
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1answer
9 views

Is this relation P an equivalence relation or a partial order relation?

I am having trouble with partial order and equivalence relations. I was wondering if someone can guide me through this problem. Let $Σ$ be the set of letters {$a, b, . . . z$}. Let $Σ^∗$ be the set ...
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0answers
22 views

Quotient Groups with symmetric $S_4$ group [duplicate]

I'm working on this problem and I am having trouble figuring it out. The problem is: Find the quotient group $G/H$. Write out the distinct elements of $G/H$ and construct a multiplication table of ...
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0answers
11 views

Help needed for statistical analysis of pitch class sets

Within Music Analysis, there is a quite mathematical type of analysis which looks at pitch class sets ($pcs$), not surprisingly known as pitch class set analysis. See ...
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1answer
21 views

Permutations and Combinations 2

The word ARGENTINA include the four consonants R,G,N,T and the vowels A,E,I How many of the arrangements have a consonant at the beginning,then a vowel,then another consonant at the beginning,then a ...
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2answers
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Permuations and Combinations problem

A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position.Three players are chosen to collect a gold ...
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1answer
22 views

indecomposable permutation

we say the permutation $\sigma = a_1 \dots a_n \in S_n$ is indecomposable if ${a_1 , \dots a_j} \neq [j] ,\forall j<n$. let f(n) be the count of the indecomposable permutations in $S_n$. show: ...
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1answer
19 views

Selecting “either representative” Permutation

A Chess club consisting of $14$ Math majors, $11$ EE majors and $11$ CS majors. In how many ways can the club select a president and vice president if either the president or the vice president must ...
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1answer
41 views

Permutations and Combinations exam question

Before I proceed with my queries I think it's best to present the question at hand. A class consisting of 4 males and 12 females in randomly divided into 4 groups of 4. What is the probability ...
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3answers
28 views

Proving $AD_1A^{-1}=D_2$

I want to prove that if $A$ is a permutation matrix, and $D_1$ is diagonal, than $AD_1A^{-1}=D_2$ where $D_2$ is also a diagonal matrix. I have worked out that $A^{-1}=A^T$ and I can see that the ...
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0answers
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Nice embedding of the permutohedron of order $n$ in ${\mathbb R}^{n-1}$

The permutohedron $P_n$ of order $n$ ($n\geqslant 2$) is the convex hull of the points $P_\pi=(\pi(1),\dots,\pi(n))$ where $\pi$ ranges over all permutations of $\{1,2,\dots,n\}$. Obviously, since ...
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2answers
38 views

Circular arrangements of identical objects [duplicate]

Q> In how many ways can 5 identical red beads, 3 identical green beads and 2 identical blue beads be arranged in a necklace?
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1answer
60 views

$a_n=a_0a_{n-1} + a_1a_{n-2} \dots + a_{n-1}a_0$ [on hold]

let σ∈Sn and τ∈Sm with m$\leq$n . We say σ avoid the permutation τ if there are no subset ${j_i< \dots <j_m} \subset [n]$ with: $\sigma(j_i)<\sigma(j_l) \Leftrightarrow \tau(i)<\tau(l)$ ...
3
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1answer
36 views

Deducing that the inverse of a permutation matrix is its transpose

I would like to verify that my proof below is sound. Let $A\in P$ where $P$ is the set of all permutation matrices (only one 1 in each row and column). Also, let $(A)_{ij}$ denote the entry of $A$ in ...
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2answers
33 views

Stuck in a problem in permutation and combination.

I am solving problems in permutation & combination and stuck in this problem. Two players $P_1$ and $P_2$ play a series of $2n$ games. Each game can result in either a win or a loss for $P_1$. ...
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0answers
26 views

A combinatorics question about selection strategies

I am given a set of balls--red and blue. In each set, there are three kinds of balls--small, medium and large. In each set there are 10 balls of each color: 10 Red balls (2 small + 3 medium + 5 ...
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1answer
19 views

How do I read this equation related to Combinations with repetitions in natural language?

Here's an Article from TopCoder about Combinatorics, that starts by introducing some basic concepts such as: Combinations and permutations. That part I understood just fine, but then the article ...
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20 views

Bit String Probability [on hold]

Given a bit string of length 8 begins with a 0, find the probability that it contains exactly three 0's. How many bit strings of length 8 contain an evan number of 0's? How can permutations and ...
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1answer
21 views

Bitstring Probability

I am not understanding how to apply n choose r and permutations to the following problem. Given a bit string of length 8 that has exactly three 0's, what is the probability that the bit string will ...
0
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2answers
32 views

How many 5 digit numbers can be formed out of {1,2,3…,9} where a digit can repeat at most twice?

The question is: How many different numbers of 5 digits can be generated out of {1,2,3,4,5,6,7,8,9} such that no digit can appear more than twice ? That is a number like 11213 is not allowed. but ...
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0answers
8 views

Prove a certain property of the Hodge double star operator

I want to solve the following problem Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ where $ \displaystyle \ast\omega =\sum_I \text{sgn}(I,J)\omega_I dx^J$ and $\omega$ is a k-form in ...
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1answer
19 views

Group theory disjoint cycles

Let $a=(1 3 5)(1 5 6)(1 3 5)$ I had to write this as a product of disjoint cycles and got $(1 5)(3 6)$ which I believe is correct. Then figure out $a^{24}$ and $a^{25}$. Now $a^{24}$ is the ...
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0answers
27 views

Number of Unique Permutations of 3 digits (-1,0,1) given a length that match a sum

Say you have a vertical game board of n length (length being number of spaces). And you have a three sided die that has the options: go forward one, go back one, and stay. If you go below or above ...
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1answer
28 views

Ways to arrange ALGEBRA so AA occurs

So the permutations of this qould be 7!, and I know that there are 2 objects of type A, but how can we isolate the events where those objects occur consecutively?
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1answer
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How to find the rank of linear permutation when replacement is allowed?

Question: If all $5*5*5*5*5*5*5*5=5^8=390625$ 8-digit numbers obtained by arranging (permuting) the five digits $2, 3, 6, 7$ & $9$ with their replacements are arranged in the correct increasing ...
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3answers
32 views

How many different possible expressions can I have?

I have three numbers $a,b$ and $c$ How many different additions can I have ? $a + a + a = 3a$ $a + a + b = 2a + b$ However, $a + b + a =2a + b$ which is the same addition as above so I neglect it. ...
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2answers
17 views

finding number of triangles inscribed in a circle

How to find the number of acute angle and obtuse angled triangles that can be inscribed in a circle containing 'N' equally spaced points.
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0answers
19 views

Order of two vectors to maximise the norm

Given vectors ${\bf a} = [a_1, \dots , a_n]^T$ and ${\bf b} = [b_1, \dots , b_n]^T$, a permutation $\pi$ acting on $[1, \dots ,n]$ and defining ${\bf b}^{\pi} = [b_{\pi(1)}, \dots , b_{\pi(n)}]^T$, ...
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0answers
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How to fill number of positions with given operators? [closed]

We have 4 position between 5 numbers ....and 3 operators (+,*,/) to fill this position... for example 1_2_10_15_25 we can have 1+2*10*15/25 or 1+2+10+15+25 (Repetition of any operator is allowed) ...
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1answer
33 views

How to multiply permutations together?

This is straight from an exam question: Find the order of the permutation $(1465732)(358)(79)$ in $S_9$ So I understand that I first have to write this permutation in disjoint cycle notation, but I'm ...
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Is the sequence generated by two permutations periodic?

It's quite easy to prove that given an application: $\sigma:[1,n]\to [1,n]$ we know that the sequence: $$Id_n,\sigma,\sigma^2,\sigma^3,\cdots,\sigma^m,\cdots $$ Is periodic after some index $k\leq ...
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4answers
43 views

Combinations and Permutations in coin tossing

I understand the formulae for combinations and permutations and that for the binomial distribution. However, I'm confused about their application to coin tossing. Consider three tosses. Outcomes ...
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0answers
34 views

Combinations or permutations

I have 3 particles and 5 energy levels (0E,1E,2E,3E,4E). I require all possible ways such that the sum of 3 particles equals 6E. Is there a formula that would enable me to compute the possible ways?
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2answers
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Combinations and Permutations - tiling a $52\times 3$ grid with $78$ dominos

A grid with $3$ rows and $52$ columns is tiled with $78$ identical $2\cdot1$ dominoes. In how many ways can this be done such that exactly two of the dominoes are vertical. Is this right?- ${78 ...
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1answer
44 views

The greatest number of points of intersection of n circles and m straight lines is-

The question is about combinatorics. I have no idea on how to start solving the problem. Please guide me. $(a) 2mn+ {m \choose 2}$ $(b) \frac{1}{2}m(m-1)+n(2m+n-1)$ $(c) {m \choose 2}+2({n \choose ...
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2
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1answer
28 views

Show $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$

Let $n \geq 2$ be an integer and $i, j \in \{1, 2, ..., n\} $ be distinct elements. Let $\sigma \in S_n$, Show that $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$ let ...
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1answer
37 views

2x2 grid game problem

A friend of mine is attempting to make a webpage that has a game for a 2x2 grid that is similar to the old North, South, East, West game. I cannot for the life of me figure this out. Essentially, ...
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1answer
18 views

What would the expected number of swaps in a merge sort be?

If I were given a list of random numbers say x1, x2, .........., xn and these numbers are sorted according to the merge sort algorithm. What would be the number of expected swaps/exchanges which would ...
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2answers
25 views

Permutations; group of 5 boys, 10 girls. What's the probability the person the 4th position is a boy?

Problem description: A group of 5 boys and 10 girls is lined up in random order -- that is, each of the 15! permutations is assumed to be equally likely. What is the probability that the person in ...
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2answers
24 views

If $m$ and $n$ women are standing toghter ]such that no men are woman are adjacent together what are the number of Permutations

suppose $m$ men and $n$ women from a single line in such a way that no two men are next to each other and no women are next to each other how many lineup are possible ? Never solved these problems ...
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1answer
41 views

Generating function of derangements

I am pretty new to the topic of generating functions and I would appreciate if someone could help me out with this problem I have. In the lecture we have proven the following generating function for ...
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Combinatorics: Password consisting of 13 characters. Must contain at least one odd digit, and at most two even digits. How many passwords?

I'm really trying here. I just need help where to go next. Each character is one of the 10 digits 0, 1, 2, ... , 9 What I have so far is that there are 10^13 possible passwords. I'd have to subtract ...
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0answers
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Finding a permutation class that has a growth rate greater than 1 and less than 0?

In a permutation class, there is an upper growth rate such that $gr(C)=\limsup_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$ and a lower growth rate such that $\liminf_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$. ...
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1answer
25 views

What did I do wrong in the permutations question.

I was given the following question: A hardware store sells numerals for house numbers. It has large quantities of the numerals 3, 5, and 8 but no other numerals. How many different house numbers, ...
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0answers
26 views

Equivalence of right and left cosets of two different subgroups.

Let $A$ and $B$ be two (not necessarily equal) abelian subgroups of $S_5$. If $x$ is an element of $S_5$, under what condition is the following satisfied $$xA = Bx$$ Update: The original question I ...
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1answer
32 views

Number of ways a multiple choice exam can be answered if no two consecutive answers are the same

How many different ways can you answer a 7- question multiple choice exam (with 3 choices) if you know that no two consecutive answers are the same?
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2answers
61 views

How many sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels?

How many sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels? I am so lost and confused, but here's my approach: ...
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0answers
18 views

Middle term and middle factor of Permutation

How to get the middle term and the middle factor of a permutation (ex: 15P7)? Also what is the difference between them?
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2answers
22 views

Is there a shortcut to this combination problem?

The question I have encountered is: From 4 oranges, 3 bananas and 2 apples, how many selections of 5 pieces of fruit can be made, taking at least 1 of each kind? So the method I used to solve this ...