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

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No of ways of making a selection.

There are $n$ different books and $p$ copies of each. Find the number of ways in which a selection can be made. My attempt: When he says, make a selection, I assumed that one book is to be ...
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1answer
46 views

To find number of questions in a test when number of wrong answers is given

In a certain test there are $n$ questions, in this test $2^{n-i}$ students gave wrong answer to at least $i$ questions where $i=1,2,3,\ldots,n$. If the total number of wrong answers given is ...
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1answer
39 views

Smallest value of n to form 900 n-digit numbers using given digits [closed]

An $n$-digit number is a positive number with exactly n digits. $900$ $n$-digit numbers are to be formed using only $2$, $5$ and $7$. What is the minimum value of $n$ for which this is possible? ...
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1answer
25 views

Ways of dividing people into groups.

The number of ways of dividing $12$ people into $3$ groups of $4$ each is? My attempt:First we choose $4$ members in $(12C4)$ ways.And then out of remaining $8$ we choose them in $(8C4)$ ways.Finally ...
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1answer
57 views

Permutations of Characters

How many strings of five characters use the letters a, b, c, d and do not have the letters bad appearing consecutively and in this order? (Letters can occur any number of times, including not at all.) ...
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18 views

Simple lemma about permutations

While doing some recalling about permutations I've crossed with the following simple lemma: Let $g:[n]\to [n]$ be a permutation. Let $x\in [n]$, and there exist $1\leq i\leq n$ so that $g^i(x)=x$. ...
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1answer
55 views

problems on permutation

There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of ...
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0answers
57 views

Elusive closed form for card permutation problem

Does a closed form formula f(n) exist for the two rightmost columns? The two question marks are meant to be 0. The diagram is a summary of the numerical results from original question: Permutations ...
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1answer
20 views

Possible Ecommerce Product, Size, and Option combinations

I have an ecommerce site where you can specify the size of a product and any number of options depending on the different products. Each option has a category. For example, for each product you can ...
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1answer
60 views

Does anybody spot anything familiar in this integer sequence?

$0,3,9,21,40,67,106,154,220,298,395,510,644,\dots$ These are the maxima of the distances between permutations of length $n$ up to $n=13$ according to a modified version of Spearman's footrule number ...
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1answer
25 views

How are the equivalence classes made of orbits defined

I am reading through Freileigh, and I'm curious to know if I am understanding the definition correctly. Each Permutation of a set $A$ determines a natural partition of $A$ into cells with the ...
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0answers
53 views

Counting all permutations(repeating) with no adjacent elements equal and m majority elements.

Counting all permutations(repeating) of 1 to n which have size n. 1. with no adjacent elements equal, and 2. m majority elements(A element is majority element when it appears max number of times in ...
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1answer
29 views

Number of permutations with subset distance constraint

The problem is to calculate the number of all unique permutations of a string with repetitions. There is also a constraint for one subset elements to be spaced from each other. Typical input data is ...
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1answer
13 views

Number of permutations on a lattice

Let's say I have a lattice like that: It's a lattice 10x10 (or N*N). So 100 little squares compose the lattice. Now I have to put 6 green squares (or n green squares) on the lattice. How do I ...
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4answers
93 views

How to visualize permutations?

I'm getting a warning that this is a subjective question, and it very well probably is. But nevertheless, it is still a valid question that helps in the studying of mathematics from my point of view. ...
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2answers
28 views

Discrete Math Combinatorics, permutation, one-to-one proof

I am having trouble getting started with the following proof: (This is homework, so I'd appreciate a nudge in the right direction.) Let m, r $\in$ N with 0 $\leq$ r $\leq$ m. Prove that the number of ...
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1answer
21 views

Method for finding permutation of n elements if you have all permutations of (n-1) elements

In the "The Art of Computer Programming Volume 1 third edition " chapter 1.2.5. Permutations and factorials professor Knuth introduces method for constructing all permutation of $n$ objects from ...
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3answers
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Proving that matrices $B$ produced by transposing two rows of $I$ satisfy $B=B^{-1}$?

For example, I have a $3 \times 3$ identity matrix. If I exchange rows $2$ and $3$, then I get $$ B = \pmatrix{1&0&0\\0&0&1\\0&1&0}. $$ In this case it can be checked that ...
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0answers
33 views

Permutations of k types for a number smaller then total of k's (without replacement)

How can you find the number of permutations of a set of items grouped in different categories when you must choose less than the total of the set. For example, I have a set of $12$ songs that are in ...
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1answer
39 views

How many inversions in a permutation?

For the permutation $\begin{pmatrix}5&2&4&3&6&1\\4&1&3&2&6&5\end{pmatrix}$, determine if it is even based on its inversions. So here's my trouble; I've worked ...
2
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2answers
30 views

Probability question on finding number of ways to label all boxes incorrectly with respect to the color of the ball they contain

Question: There are balls of 5 different colors - yellow, blue, red, green and white. A worker has to separate these balls as per their colors into different boxes and label them with corresponding ...
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0answers
81 views

permutations of length N of N natural numbers from 1,2,3…N such that no two consecutive numbers are equal.

For given n, I need to find sum of count of most occurring number for each permutation of length n.(such that no two consecutive numbers are equal). for e.g. for n=3 1 2 3 will give 3 as the count ...
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1answer
37 views

Transposition as generating $S_n$.

As far as I know the group of symmetries of the euclidean n-cube is generated by reversions ${r_i}$ and transpositions ${s_i}$ for i=1,...,n-1. Transpositions generate the subgroup of permutations of ...
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2answers
52 views

the number of inversions in the permutation “reverse”

Known, that number of inversions is $k$ in permutation: $$\begin{pmatrix} 1& ...& n& \\ a_1& ...& a_n& \end{pmatrix}$$ Find number of inversions in permutation (let's ...
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3answers
45 views

Number of combinations of k types from set n with limited number of each type

How can I find the number of combinations of types from a set of multiple types when the number of each type is limited? For example, I have a set of 3 chicken dishes, 4 beef dishes, and 5 lamb ...
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Combinatorics : number of non-decreasing series of r distinct numbers where the size of series ranges from 1 to N

I understand that number of non decreasing sequences of size M with N distinct numbers is (N+M−1)C(M). However, I'm interested in finding out the number of such series of r distinct integers where the ...
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1answer
26 views

Determine if a lottery system is profitable

I have a problem where I'm supposed to determine if a lottery system is profitable. I solved the problem and found it to be profitable, but I am not 100% sure about all of my calculations. Below is ...
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2answers
71 views

In how many ways can you distribute $3$ chocolates among $2$ kids if you have to give all $3$ of the chocolates to the kids? [closed]

In how many ways can you distribute $3$ chocolates among $2$ kids? One kid can get none. But we need to give away all $3$ of the chocolates to the $2$ kids. Why is it wrong to use $3C2$(which gives ...
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2answers
51 views

Normalizer of the cyclic group in $S_n$

Let $G = S_n$ and $H = \langle (1,2,\ldots,n) \rangle.$ It is not too hard to see that $$C_G(H) = H.$$ What I am now wondering is, which group is $N_G(H)?$ Is there any way to determine that? I ...
2
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2answers
74 views

Expected value of random permutation

Let n ≥ 1 be an integer and consider a uniformly random permutation $a_1$, $a_2$, . . . ,$a_n$ of the set {1, 2, . . . , n}. Define the random variable X to be the number of indices i for which 1 ≤ i ...
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1answer
36 views

Cyclic type of the product of two permutations

Maybe this question is way too simple but I'm stuck. Suppose $x,y \in S_n$ and the product $xy$ has the cyclic type $(t_1,t_2,\ldots,t_n)$; here $t_i$ is the number of cycles of length $i$ when we ...
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1answer
14 views

permutations repetitions/no repetitions

License plates consist of sequence of 3 letters followed by 3 digits. How can they be arranged if (i) no repetition of letters is permitted, how many possible license plates are there? should it be ...
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1answer
44 views

Permutations : If repetitions are allowed

For example if a question is to find the number of different ways of arranging $4$ letters of $26$-letter alphabet with repetition, I know that we have to do $26^4$. However, I am confused as to why ...
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1answer
36 views

Number of combinations in a string with n states

I have a problem in biology involving amino acids (think of them as a string of characters) that I want to formalise. Let assume we have a amino acid sequence of length 4, typical examples may be: ...
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Cycle structure of affine transformation

Consider the ring $\mathbb{Z}_n$ of remainders modulo $n$ for some number $n.$ Let $a,b \in \mathbb{Z}_n$ and consider the map $$f_{a,b}(x) = ax+b.$$ If $a$ is invertible then the above map is ...
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1answer
37 views

How many cycles $A$ and $B$ can form this cycle

How many cycles $A$ and $B$ can form this cycle: $AB=(axyguimjrcwk)(bvqphsleofzt)(d)(n)$ I can see that $A$ and $B$ must share the cycle $(dn)$, and I believe due to ordering, both $A$ and $B$ must ...
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1answer
28 views

$H$-orbits in X have not the same cardinality if $H$ is not normal in $G$

Let $G$ be a transitive subgroup of the symmetric group $S_n$ on $n$ letters, and let $H$ be a normal subgroup of $G$. I know that the action of $G$ on the set $X =\{ 1,..., n \}$ induces a natural ...
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2answers
14 views

consecutive combination of n things taken k

Let's start my question with a simple example. Suppose I have $4$ apples that are numbered 1 to 4 and I want to to choose $2$ of them. The first and easy answer is $4C2$ but I want them to be ...
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3answers
69 views

Prove that $Z(S_n) = \{(1)\}$ for every $n \geq 3$. Induction

I wonder if this questions can be done by induction. $S_3 = \{(1),(12),(13),(23),(123),(132)\}$ $Z(S_3)$ contains all the elements in $S_3$ that commutes with all the element in $S_3$ We can easily ...
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2answers
25 views

Writing permutations as products of disjoint cycles

How can I write these permutations as products of disjoint cycles? i.$\;\;(1234)(513)$ ii.$\;\;(13526)(53)(46215)$ iii.$\;(13)(12)(32)(143)$
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Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
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1answer
23 views

Problem in permutation groups involving conjugates

I have to find a permutation $a$ satisfying $ a xa^{-1}=y$ where $ x=(12) (34)$ and $y=(56) (13)$ My attempt in solving the problem was- $$ a(12)(34)a^{-1}= a(12)(a^{-1}a)(34)a^{-1}= ...
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1answer
31 views

a problem involving permutation groups

I am given two cycles $(123)$ and $(456)$ and have to find a permutation $ \sigma$ (if it exists) such that $ \sigma(123) \sigma^{-1} = (456)$. This is what I tried: let $ \sigma(123) \sigma^{-1} ...
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1answer
36 views

Probability of an event if items are drawn simultaneously or not

A box contains 10 balls, with each ball labeled from 1 to 5 (there are two balls labeled with a 1, two labeled with a 2, and so on). 3 are drawn without replacement. What is the probability of drawing ...
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2answers
75 views

Counting all possibilities that contain a substring

How many strings are there of seven lowercase letters that have the substring tr in them? So I am having a little problem with this question, I know that the total number of combinations is $26^6$ ...
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2answers
34 views

Take seven courses out of 20 with requirement

To fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of 7 courses must be a statistics course. ...
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0answers
56 views

Am I over counting?

Two chess players, A and B are going to play 7 games. Each game has three possible outcomes: a win for A (which is a loss for B), a draw (tie), and a loss for A (which is a win for B). A win is ...
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2answers
54 views

What is the answer to this P&C problem.

What is the answer to the below mentioned P&C problem: BurgerTown offers many options for customizing a burger. There are 3 types of meats and 7 condiments: lettuce, tomatoes, pickles, ...
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0answers
37 views

All non-isomorphic transitive actions of the Dihedral group

Consider the Dihedral group $D_n$ of order $2n$ as a permutation group. That is $$D_n = \langle (1,2,\ldots, n), (1)(2, n-1)(3,n-2) \cdots \rangle.$$ I would like to determine all faithful transitive ...
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1answer
39 views

Permutations: Discrete Math

How many permutations are there of the set $(a,b,c,d,e,f,g)$ My Answer: Since there are 7 elements in the set, $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$ Am I right?