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

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How many permutations of $x^5 + y^5 + z^5$ are possible given x, y, z are integers such that $1 \le x \le y \le z \le 180$?

I initially thought it would be $180^3$ possible permutations, but then quickly realized that something like $x=3, y=2, z=1$ would not be valid due to the constraints. How can I go about trying to ...
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1answer
21 views

How to solve this equation containing $P(n,r$)?

$P(n,r) := \frac{n!}{ (n-r)!}$ The equation is: $P(n,r) = 42 P(n,3)$ I need to clear the variable $n$. It doesn't matter that it has to be expressed as a function of $r$. I cannot pass the step: ...
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1answer
21 views

Arrangement of the word MATHEMATICS if last spot must have the letter 'T'

How many ways can the word MATHEMATICS be arranged if the last letter must be a T? My solution: There are $2$ possible choices for the last letter (There are $2$ different T's), which leaves $10$ ...
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1answer
21 views

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
24 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
11 views

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
20 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
43 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 ...
2
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0answers
10 views

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
40 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: $\sigma(j_i)<\sigma(j_l) \Leftrightarrow \tau(i)<\tau(l)$ ...
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1answer
37 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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0answers
21 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
35 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
28 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
11 views

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
17 views

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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0answers
27 views

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
35 views

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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0answers
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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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0answers
28 views

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
10 views

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?