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

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Classes of $E=\{ 1,2,3 \}$

Question $2$: If $E=\{ 1,2,3 \}$ Determine the classes of all $E$ elements of E for each permutation $E=\{ 1,2,3 \}, \quad \Omega_{\sigma}(x)=\{ \sigma^{m};m\in \mathbb{Z} \} $ My Thoughts ...
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1answer
22 views

Is it a correct equation for permutations with sets of indistinguishable objects?

C(n, r) = P(n, r)!/r! = n!/r!ㆍ(n-r)! I'll check if the right hand side of the above equation in Theorem 9.5.2 is correct by expanding the left hand side. $C(n, n_1)ㆍC(n-n_1, n_2)ㆍC(n-n_1-n_2, ...
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21 views

Does a function between sets induce a homomorphism between the respective permutation groups?

Let $X,Y$ be finite sets, and let $\Sigma(X),\Sigma(Y)$ be their respective permutation groups. Consider a function $f:X\to Y$. Is there a homomorphism $\phi:\Sigma(X)\to\Sigma(Y)$ induced ...
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2answers
65 views

From a pack of 52 playing cards, three are drawn at random. Find the probability of drawing a king, a queen and jack.

A simple question but the solution is confusing me. The answer I obtained was $$p = 3! \times 4/52 \times 4/51 \times 4/50$$ The first 3! is for the order of king, queen, jack. $4/52$ is the ...
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1answer
66 views

Show that there is no subgroup of $S_n$ of order $(n-1)!/n$.

I am trying to show that $S_n$ does not have a subgroup of order $(n-1)!/n$ for any $n$ other than $6$. I have checked it to be true up to $S_{13}$. Any ideas? Of course, if $n$ is prime then that ...
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2answers
50 views

Find the number of ways of distributing 15 chocolates among 3 kids where none gets less than 3 and more than 6.

Find the number of ways of distributing $15$ chocolates among $3$ kids where none gets less than $3$ and more than $6$. Now, $a+3+b+3+c+3 = 15$. $a+b+c=6$. Number of ways of distribution $= (6+3-1) ...
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1answer
87 views

How many strings possible with atmost 2 distance away?

So my question is you have given a string of n length , say abcd (n=4) . so how many unique strings of same length (n=4) possible such that they become at most 2 distance away from given string ? ...
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3answers
29 views

Counting - Permutations & Combinations

Two series of a question booklet for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each so that there should be no identical ...
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35 views

Number of ways to collect ticket

There are 5 people with \$$5$ each and $5$ people with \$10 each such that they are in a bus stop.A ticket collector has to collect tickets from them and he has no change initially.The ticket ...
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4answers
53 views

How many entries in a symmetric matrix can be chosen independently?

If $B \in \mathbb{M}_n (\mathbb{R})$ is a symmetric matrix (i.e. $B = B^T$), how many entries of B can be chosen independently.
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27 views

On cycle decomposition in $S_n$.

Pick random $\pi,\tau\in S_n$ and denote $\sigma=\tau\circ\pi$. What is the probability in cycle decomposition $\sigma=c_1\cdot\dots\cdot c_r$ there is no $c_i$ with length larger than $f(n)$ for ...
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1answer
258 views

Number of pairs of strings satisfying the given condition

We are provided with a string A (may contain repeated characters as well ). One needs to make all possible permutations of A , and find out how many pairs of strings chosen from these permutations are ...
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1answer
20 views

Why reflection and rotation are sufficient operations in dihedral group?

I know a bit of elementary group theory but please ignore dihedral group in the title and let's make it simple enough so a high student can read the question and the answer(s)... Suppose we have a ...
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0answers
55 views

At what rate does the entropy of shuffled cards converge?

Consider a somewhat primitive method of shuffling a stack of $n$ cards: In every step, take the top card and insert it at a uniformly randomly selected one of the $n$ possible positions above, between ...
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2answers
43 views

Prove that exactly half of the permutations in $S_n$ have a negative signature

If $S_n=\lbrace \alpha : \mathbb{Z}_n \rightarrow \mathbb{Z}_n \mid \alpha \text{ is one-to-one and onto}\rbrace$ is the symmetric group on $n$ letters, that is, $S_n $ consists of all ...
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5answers
126 views

proving that $\frac{(n^2)!}{(n!)^n}$ is an integer

How to prove that $$\frac{(n^2)!}{(n!)^n}$$ is always a positive integer when n is also a positive integer. NOTE i want to prove it without induction. I just cancelled $n!$ and split term which are ...
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1answer
34 views

Total number of non similar triangles which can be formed such that all the angles of the triangles are integers

My question is: " Find the total number of non similar triangles which can be formed such that all the angles of the triangles are integers" My attempt: Let $x$, $y$ and $z$ be the angles of the ...
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4answers
60 views

Number of ways that can 12 persons take seats in a row of 20 fixed seats so that every person has exactly one neighbour

I am not able to take this question.My question is "Find the number of ways that can 12 persons take seats in a row of 20 fixed seats so that every person has exactly one neighbour" My Attempt: I ...
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1answer
47 views

How to select an item from a set with constraints, and finding when the pattern repeats.

I am trying to solve the following problem with constraints that are explained below. Assume there are customers walking into a restaurant and you want to assign them to different waiters. With the ...
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1answer
133 views

Let $\sigma\in S_n$ be a $k$-cycle, $k>1$. Show that $\sigma^j$ (where $j$ is an integer) is a cycle if and only if $j$ is coprime with $k$

Let $\sigma \in S_n$ be a $k$-cycle, $k>1$. Show that $\sigma^j$ (where $j$ is an integer) is a cycle if and only if $j$ is coprime with $k$.
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1answer
42 views

How many ways can a selection be made?

Suppose a company will select 3 people from a collection of 12 applicants to serve as a regional manager, an assistant regional manager, and an assistant to the regional manager. In how many ways can ...
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0answers
49 views

Abstract algebra: for a polynomial $p$, prove $\sigma(\tau(p))=(\sigma\tau)(p)$ for all $\sigma, \tau \in S_n$

I'm trying to solve to following problem: Part 1: Let $\{x_1,...x_n\}$ be variables. For any polynomial $p$ in $n$ variables and for $\sigma$ $\in S_n$ define $\sigma ...
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1answer
46 views

How many ways are there to arrange the letters in the word “mississippi” such that all “p” precede all “i”?

How many ways are there to arrange the letters in the word "mississippi" such that all "p" precedes all "i"? My possible solution: Consider: p p _ _ _ _ _ _ _ _ _ so 9! ways to arrange words(?) ...
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1answer
86 views

How many permutations of the letters TRIANGLE contain A, N and G not to be together ?

I've approached this by calculating $8!$ for the total permutations, and subtracting $6!$ (permutations that include ANG.) $8! = 40320$ $6! = 720$ Thus, $8! - 6! = 39600$ Is this the correct ...
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4answers
75 views

How many rectangles can be obtained by joining four of the $12$ vertices of a $12$-sided regular polygon?

What is the number of rectangles that can be obtained by joining four of the 12 vertices of a 12-sided regular polygon? The thing that I've tried is to find out at what intervals should I connect to ...
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1answer
19 views

Calculate Number of Trajectories

An object moves on a coordinate plane from point (0,0) to point (15,11) in a series of steps, where each step increments one of the two coordinates. How many different trajectories are ...
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1answer
34 views

Let f be an r-cycle in $S_n$. Given any $h \in S_n$, show that $hfh^{-1} = (h(x_1), h(x_2), \dots, h(x_r))$

The problem: let $(x_1, x_2, ..., x_r)$ be an r-cycle in $S_n$. Show that for every $h \in S_n$, $h \circ (x_1, x_2, \ldots, x_r) \circ h^{-1} = (h(x_1), h(x_2), \ldots, h(x_r))$
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1answer
13 views

For which $k$ I could sort permutation?

Suppose we have some permutation: $$p = a_{1}..a_{n}$$ We could inverse some subarray length of $k$. So my question is: for which $k$ I could sort my permutation using only my $k$-inverse? Obviously ...
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56 views

Cardinality of a group of permutation

Let $S$ be an infinite set of cardinality $\alpha$ and $G$ be a subgroup of $Sym(S)$. Let $\sigma(g)=\{s\in S \mid sg\neq s\}$ for each $g\in G$ and define $$Sym(S,\, \alpha)=\{g\in Sym(S)\mid ...
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2answers
67 views

Coupon Collector Problem with packs

Given the coupon collector's problem, the expected number of coupons is calculated as follows: $E[X] = N \sum_{i=1}^N \frac{1}{i}$ This assumes we can draw one coupon at a time. Let's assume one ...
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1answer
42 views

When the product of any two consecutive digits in the number is a prime number

I came across a question today. The number of 10-digit numbers such that the product of any two consecutive digits in the number is a prime number, is? As much as I know the product of any two ...
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0answers
31 views

If $G = R \rtimes G_{\alpha}$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \ncong D_{12}$

Suppose that $G = R \rtimes G_{\alpha}$ is a finite permutation group on $\Omega$ where $R$ is the non-abelian subgroup of order $27$ and exponent $3$ (see here). Suppose that $G_{\alpha} \cong D_n$, ...
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47 views

Multinomial combinations with inequalities

The number of ways you can arrange $N$ balls into $m$ bins such that each bin has $\{n_1, n_2,...,n_m\}$ balls ($n_1+n_2+...+n_m=N$) is the standard multinomial $$\frac{N!}{n_1!n_2! \dots n_m!}$$ ...
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3answers
64 views

In how many ways $7$ pencils can be distributed to $5$ children such that each child can get any number of pencils?

Q - In how many ways $7$ pencils can be distributed to $5$ children such that each child can get any number of pencils? I am little confused whether the answer is $5^7$ or $7^5$ ? I think every ...
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1answer
17 views

Permutation multiplication of non-disjoint cycles in $S_4$

I am having an issue with calculating the product of permutation cycles for calculating commutators. Consider $A_4$ and the commutator $[(123),(14)(23)]$ So we have $[(123),(14)(23)] = ...
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1answer
24 views

How to find explicit form of recurrence relation with four variables for combinatorical value

I want to know how many ways there are to choose $l$ elements in order from a set with $d$ elements, allowing repetition, such that no element appears more than $3$ times. I've thought of the ...
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1answer
44 views

Is the cyclic permutation $(1 2 3)$ equal to $(12)$ followed by $(13)$ or $(13)$ followed by $(12)$

At the bottom of this page: http://dogschool.tripod.com/permgroups.html it states that (1 2 3) This is equivalent to two transpositions: (1 2) followed by (1 3) [try it!] So I did try it: ( 1 2 ...
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2answers
47 views

1) In how many ways can letters of the word $\text{MAMMAL}$ be arranged in a line?

1) In how many ways can letters of the word MAMMAL be arranged in a line? For this question I put $$\frac{6!}{3!2!} = 60$$ since there are $3$ 'M's and $2$ 'A's 2) The letters of the word ...
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2answers
40 views

In how many different ways can $3$ red, $4$ yellow and $2$ blue bulbs be arranged in a row?

$1)$ How many different ways can $3$ red, $4$ yellow and $2$ blue bulbs be arranged in a row? Do I just say $3! 4! 2! = 288$ ? $2)$ On a shelf there are $4$ different math books and $8$ different ...
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5answers
66 views

Prove that $\frac{(n+1)!}{(n-1)!}=n(n+1)$

Simplify $$\frac{(n+1)!}{(n-1)!}$$ My book shows the answer as $n(n+1)$. I don't know how does it come up? I have tried: $$\frac{(n+1)!}{(n-1)!}=\frac{(n+1)n!}{(n-1)(n-2)...n!}$$ I have been ...
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0answers
41 views

Compressing permutations

I am looking at a compression technique that finds a permutation of the bits of a 64-bit block of data that collects all 1 bits at the beginning of the block, and sends all 0 bits to the end of the ...
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1answer
30 views

Vertices of Cube coloured with black or white colour

In how many rotionally distinct ways can the vertices of a cube be coloured with black or white colour ? I don't know how to approach this question. Please provide some insight.
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1answer
44 views

Probability: Letter Arrangements

In how many ways can the letters of the word ARRANGEMENTS be arranged? a) Find the probability that an arrangement chosen at random begins with the letters EE. b) Find the probability that the ...
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2answers
39 views

Rectangular Table Arrangement

a) In how many ways can 13 people be seated on one side of a rectangular table if Doug refuses to sit next to Gordon? I have two different ideas- Idea 1) There are two options: either Doug is at ...
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3answers
72 views

Find the number of distinct integers with non-decreasing digits formed from one or more of the digits $2, 2, 3, 3, 4, 5, 5, 5, 6, 7$

Suppose integers are formed by taking one or more digits from the following: $2, 2, 3, 3, 4, 5, 5, 5, 6, 7$. For example, $355$ is a possible choice while $44$ is not. Find the number of distinct ...
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1answer
48 views

If $|\alpha^N| = 1 + k|g|$ for $g \in G_{\alpha}$, then $g$ fixes a point on every $N$-orbit that it stabilizers

Let $G$ be a finite transitive permutation group, suppose $p$ does not divide $G_{\alpha}$ and that $1 \ne P \unlhd G$ is a normal $p$-subgroup of $G$. Let $\Delta := \alpha^P$ be an orbit of $P$ and ...
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0answers
20 views

Number of $4\times3$ matrices of rank 3 over a field with 3 elements.

I am finding number of $4\times3$ matrices of rank 3 over a field with 3 elements. If i count it as number of linearly independent columns i.e $3$ then its answer is $(3^{4}-1)(3^{4}-3)(3^{4}-3^{2}).$ ...
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0answers
41 views

How can I find the elements generating a group in a special way?

Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le ...
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1answer
56 views

prove that $\det X(\pi) = \operatorname{sgn}(\pi)$ for all $\pi \in S_n$.

Let $X:S_n → GL_n(\mathbb{R})$ be the defining representation of $S_n$. Prove that $\det X(\pi) = \operatorname{sgn}(\pi)$ for all $\pi \in S_n$. attempt: I was thinking in trying to use $X(e) = ...
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1answer
26 views

In how many ways can 5 chocolates be chosen from an unlimited number of Cadbury,Five-star, and Perk chololates?

This might be trivial but I am still confused about it. I am having some trouble in interpreting this question. My approach was to simply convert the question into the equation $$a + b + c = 5$$ ...