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

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Number of ways to color N objects in X colors where there is at least one object of each color.

What is the number of way to color $N$ objects in $X$ colors, where there is at least 1 object of each color?
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1answer
104 views

Combination or Permutation

I am searching for the number of uniques ways to paint an icosahedron. However, my understanding of mathematics is quite limited in the field of combination and permutation. I have searched through ...
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0answers
15 views

Generating uniform permutations by a particular method

Let $A$ be a uniformly random permutation of the numbers $\{1,2,\cdots,n\}$. I want to generate a uniformly random permutation from $A$ on the numbers $\{1,2,\cdots,n,n+1,\cdots,n+m\}$. In other ...
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1answer
48 views

How many cycles of length $k$ in $S_n$?

In the symmetrical group $S_n$, how many cycles of length $k$ can we form? After some research I am tempted to say $\frac{n!}k$ but I am not sure.
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29 views

Dependent permutations, a question.

I cant seem to find anything on the internet on this subject , and the professor did not explain it too well, in short the following is unclear to me how is $$(1 3 4)(236)=(24136)$$and ...
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1answer
32 views

Symmetries of the regular hexagon

Q- Let G be the group of the symmetries of the regular hexagon. List the elements of G (there are 12 of them), then write the table of G. So for the listing the elements of G, they want it like this: ...
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0answers
43 views

Integrating over a symmetric-group function (elements being permutations)

I would like to integrate a permutation of a function. Namely I have the following: $\sum_{\sigma, \sigma'\in S_{n+1}}\int_{-A}^A dz_1dz_2 ... dz_{n+1} ...
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31 views

Ball-of-wacks combinations

The six-color version of the ball-of-wacks consists of thirty rhomboidal pieces, which can be combined to form a rhombic triacontahedron. There are six colors, each with five pieces. One challenge ...
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2answers
30 views

Illegal permutations give a nonzero answer

I am told that a random variable can take a value of $+1$ or $-1$. I am given the total number of times the random variable is counted, $N$, and the sum of the random variables, $n$, and asked to find ...
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1answer
47 views

List the elements of the cyclic subgroup of $S_6$

List the elements of the cyclic subgroup of $S_6$ generated by: \begin{smallmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 1 & 6 & 5 \end{smallmatrix} ...
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1answer
54 views

How many three digit number can be formed?

Question: (a) How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once? (b) How many of these are odd numbers? (c) How many are greater ...
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22 views

Birkhof representation of a stochastic matrix

From Birkhof Theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices although this representation might not be unique. I have the ...
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1answer
34 views

no of possible ways [duplicate]

we have to build a houses on $m$ plots, such that no two consecutive plots exist on which it is allowed to build house calculate the number of possible ways of assigning free plots to buildings ...
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1answer
45 views

The order of a $k$-cycle in $S_n$ is $k$.

Here's what I have right now: The order of a $k$-cycle in $S_n$ is $k$. Proof. Let $\sigma$ represent the $k$-cycle $$\sigma=(x_1 \ x_2 \ \cdots \ x_k)$$with distinct elements $x_i$. Note that the ...
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1answer
23 views

How to solve this statistics problem? [closed]

Can you find the sum of all numbers that can be formed with the digits $2, 3, 4$ and $5$ taken all at a time? (So its like you sum up the number from 1st digit to 4th digit) I'm learning now about ...
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2answers
24 views

Permutation's decomposition into transpositions

Transposition is a cycle with 2 elements. Any permutation can be decomposed into a product of transpositions. For example, for permutation $\begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 3 ...
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1answer
42 views

Maximize $a_1^{a_2^{\ldots^{a_n}}}$, where $(a_1,a_2,\ldots,a_n)$ is a permutation of $(b_1,b_2,\ldots,b_n)$

You are given a tuple of integers $B=(b_1,b_2,\ldots,b_n)$. Find $(a_1,a_2,\ldots,a_n)$ - a permutation of $(b_1,b_2,\ldots,b_n)$ - that maximizes $a_1^{a_2^{\ldots^{a_n}}}$. For example - If ...
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0answers
48 views

Permutations Without Repetitions

Given the set [A,B,C,D] how many distinct ways can I order all four of the members of the set? I see distinct, as a unique set, therefore [A,B,C,D] and [D,C,B,A] ...
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1answer
72 views

How to calculate combinations of multiple variables which can assume multiple values

I have 3 variables (A,B,C); each variable can assume 3 different values (1,2,3) . I want to calculate ho many combinations there are which follow this rule: let's fix A1, then cycle on all the others ...
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3answers
280 views

How many ways to reach a given tennis-score?

Let's say a tennis player wins a set with a game score of 6-3. In how many different ways can we reach this score? Assuming H means the home-player won the game and A means the away-player won the ...
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2answers
75 views

Number of Words with two letters $a$ and $b$.

Given $N$ and $M$, find the number of $N$ letter words consisting of only $a$ or $b$, where $b$ must not be consecutive for more than or equal to $M$ times. Example: if $N=3$ and $M=2$, then all the ...
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1answer
27 views

Combinations - Permutations

Find the number of ways in which 5 books can be distributed between three people A,B and C, if the books are a)indistinguishable, b)all different. a) $\displaystyle \frac{5!}{3!(5-3)!} = 10$ b)$ ...
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1answer
212 views

Counting elements in cartesian power with plurality + pattern constraints

Problem: I have an alphabet with n=8 letters (say $X=\{A, B, C, D, E, F, G, H\}$). I'm looking for words with m=24 letters, with three constraints: letter $A$ is the relative majority (like in ...
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1answer
34 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
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2answers
65 views

If there is a bijection $f: X\rightarrow Y$, prove that there exists an isomorphism $\phi :S_X\rightarrow S_Y$

If there is a bijection $f: X\rightarrow Y$, prove that there exists an isomorphism $\phi :S_X\rightarrow S_Y$. Here $S_X$ denotes the group of all permutations of $X$, i.e., the bijections $X\to ...
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1answer
30 views

Permutations and Counting problem?

Postal codes in Canada have six characters with alternating letters and digits in the form L#L#L#. How many postal codes do not have one letter repeated three times? What I did is ...
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0answers
21 views

Sampling with/without order

Our professor have presented this simple example in the lecture. You have $P_n$ processors and $M_k$ memory where $k>n$. If two or more requests goes to same memory then the request will be ...
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0answers
41 views

The $8$-Puzzle and $2$-Cycles

I have been studying the $8$-puzzle and have thus far managed to wrap my head around the following information: The following illustrates the solved position of the $8$-puzzle, where $9$ is the empty ...
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1answer
49 views

Prove that this number is an integer

Prove that the number $${4155 \cdot4156 \cdot\ldots \cdot4251 \over 2 \cdot3 \cdot\ldots \cdot 97}$$ is an integer. How might I prove this?
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1answer
45 views

Adding numbers in a consecutive series

I have the series: 1, 13, 133, 1333 ... Currently I have distributed it down to: 1 + (10 * 2) + (100 * 2) ... Can anyone point me in the right direction? Sorry, I forgot to mention, I'm looking for ...
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1answer
106 views

How many permutations of letters ABCDEFG contain the strings ABC and CDE

For this problem, I understand how to find something like how many strings contain the string BA and GF. I just look at the set of letters like this: {BA, GF, C, D, E} and since I have 5 distinct ...
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2answers
17 views

Counting permutations of a set that doesn't fix elements

I want to know how to count the number of permutations of a finite set that doesn't fix elements, i.e., the cardinality of the set $H=\{f\in S_n: f(i)\not = i\ \mbox{for}\ 1\leq i \leq n\}$, where ...
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1answer
44 views

Odd and Even Permutations and their parities

So the question is: Let Alpha and Beta belong to Sn. Prove that BetaAlphaBeta and Alpha are both even or both odd. I'm not sure where to start. My basic logic class tells me to go with the idea: ...
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1answer
49 views

Group theory, order with permutation of Z

Let $S_{Z}$ be the set of permutations of ${Z}$ (note that this is an infinite group!). Find two elements of $S_Z$ which both have finite order, but whose product has infinite order. I just am really ...
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1answer
32 views

If σ is a cycle of length n, then σ^r is also a cycle if and only if n and r are relatively prime

If σ=(1.2.3), σσσ= identity permutation, which is cyclic in this case n=3 and r=3 but their gcd is not 1. I don't understand why -> this direction of theorem is true
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1answer
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Discrete Math Boys and Girls

Problem 4: Boys and Girls Consider a set of m boys and n girls. A group is called homogeneous if it consists of all boys or all girls. In the following questions, practice the multiplication and the ...
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Demonstrate the isomorphism $\text{Perm}(X) \to S_n$ where $X$ is a finite set.

Logically, the following is extremely intuitive. However, I am having trouble expressing this in provable form: Suppose $X$ is a finite set, with $|X|=n$. Show that, given any choice of labeling ...
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29 views

Show that group actions are the same as homomorphisms $G \to \text{Perm}(X)$

The following is a homework problem. The conclusions are extremely intuitive and easy to see, but I am having proving this. Could someone please help? Show that, given a group action $G \times X \to ...
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1answer
26 views

Find the number of bit strings which start with four zeroes and end with three ones

Count the number of bit strings that start with four $0$'s or end with three $1$'s if the length of the bit string is: $7$ $4$
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2answers
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Combinations and Permutations: 3 letters, repeat exactly three times

If I have three letters, A, B, and C, and I want to use each letter exactly three times (9 places), what is the probability that I randomly pull out a nine-string of letters that starts with ABC? I ...
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1answer
32 views

Alternative answer to choose $k$ people out of $n$, then choose 1?

The question is fairly simple, we first choose $k$ people out of $n$, that is $C(k,n)$ as the combination function, then we choose 1 person out of $k$, we have $k$ choices. The total number of choice ...
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4answers
573 views

Combinations: How many handshakes?

I need help with the following question which I cannot seem to solve: 17 students are sitting in a circle. Each person shakes hands with everyone but his/her neighbours. How many handshakes have been ...
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1answer
38 views

How many proper nontrivial subgroups do D5 have?

Do I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups
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1answer
21 views

Find all nonnegative integers n, that satisfy the equation P(n, 2) = P(4n − 6, 1)

I have the equation P(n, 2) = P(4n - 6, 1) and I need to find all non negative integers which satisfy it. I understand that this equation can also be written as n!/(n-2)! = (4n-6)!/(4n-5)! and ...
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29 views

formula to get combinations of notes that can be played over chords.

I was experimenting with chord tones,and asked myself-if i'm allowed to play only one note everytime the chord changes,i would come up with a situation in which there are some fixed positions,the ...
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1answer
35 views

Express how many ways you can select a representative

Assume that a school has these three teams: Chess team with 10 members, Checkers team with 15 members, and College bowl team with 20 members. In how many ways can we select representatives of the ...
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1answer
29 views

Classifying 1 cycle permutation matrices

Given a permutation matrix that is not full rank, is there an algebraic criterion to tell if matrix contains more than one disjoint non-trivial cycle or exactly one non-trivial cycle? Example: ...
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1answer
32 views

Cycle structures of $S_6$

A problem from my algebra homework requests the following: List all the possible cycle structures in $S_6$. For each cycle structure, compute the order of an element with that cycle structure. ...
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2answers
84 views

Prove that any element in $S_n$ can be written as a finite product of the following permutations

Prove that any element in $S_n$ can be written as a finite product of the following permutations: $(a)\ (12),(13), . . . , (1n)$ $(b)\ (12),(23),...,(n−1,n)$ $(c)\ (12),(12\dots n)$. I have no ...
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2answers
61 views

Is $A_n$ non-abelian for $n= 3$?

In the book, it is asked to show that $A_n$ is non-abelian for $n ≥ 4$. Which may imply that it is abelian for $n=3$. Is that so? because $(13)(12)\ne (12)(13)$. Hence is it true to write: $A_n$ is ...