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

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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. I tried- Both the ...
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1answer
65 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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1answer
30 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
46 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
21 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
41 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
25 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
44 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
11 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
26 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
29 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
41 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
69 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
22 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
27 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 ...
2
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1answer
26 views

Possibilities of license plates with special rules

I have looked all over the web for some additional information on this matter with no results. Lets say a new form of license plate have 4 letters followed by 3 digits and all sequences are possible. ...
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1answer
14 views

Inversions and Multiplicativity of the Sign of a Permutation

The question is mainly about showing, for two permutations $\sigma, \pi \in S_{n}$, that $\mathrm{sgn}(\sigma \pi) = \mathrm{sgn}(\sigma) \mathrm{sgn}(\pi)$ using inversions of permutations (i.e. a ...
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0answers
36 views

Let $f \in 8^8$ where the permutation is given in two line form:

I'm having trouble understanding how one would answer question's like this. $$\begin{pmatrix} 1&2&3&4&5&6&7&8\\ 3&4&5&8&1&2&7&6\end{pmatrix}$$ ...
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2answers
44 views

Finding probability of intersection of events

I was reading First course in Probability by Sheldon Ross and am stuck at the understanding this simple problem [hence proved my maths is poor :( ]. Problem: Celine is undecided as to whether to ...
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4answers
164 views

Number of ways you can form pairs with a group of people when certain people cannot be paired with each other.

Let's say you have a group of eight people and you want to form them into pairs for group projects. There are $\frac{8!}{4!.2!}$ ways to do it. ($8!$ is the total number of ways $8$ people can be ...
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0answers
17 views

Counting permutations of length n without patterns

Count the number of permutations of length n that avoid patterns of high-low-mid. A pattern of hi-lo-mid is 3 integers in the pattern such that for $a_i$,$a_j$,$a_k$, we have i < j < k, $a_i$ > ...
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2answers
45 views

Probability of tossing five coins and getting at least one head

here is my dilemma. I want to know the probability of getting at least one head in five coins being tossed one after the other. Could you help me get the logic of this as it involves both mutually ...
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1answer
48 views

The number of ways of dividing a number by three separate integers.

How many ways can I arrive at the number $45$ by exactly using $5$, $10$ and $20$. I can use each number as many times as necessary. (e.g $9×5$, $20+(5×5)$) this leads to the question, if the number ...
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1answer
42 views

How to interpret combination and permutation problems?

This is more of a methods question than asking for a specific answer: In revisiting statistics and attempting various problems, I am curious if anyone has any insights on how to "see" the route to ...
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1answer
21 views

Multiple Group Representations using Cayley's Thm

I know that an abstract group can be made isomorphic to a subgroup of a symmetric group, by using a Cayley table for that abstract group. However, what is a technique for getting another permutation ...
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2answers
38 views

maths permutation help

An experiment consists of randomly rearranging the 9 letters of the word TARANTULA, where all possible orders of the 9 letters are equally likely. Find the probability of each of the following events: ...
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1answer
29 views

Permutations and group acts

How many ordered pairs of permutations $(\pi , \sigma )$ in $S_n$ such that $\pi \circ \sigma =\sigma \circ \pi $. I think i need consider group acts on itself by conjugation $\pi (\sigma )=\pi \circ ...
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1answer
45 views

Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states: Let $G$ act ...
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1answer
123 views

problem on permutations

In $S_{10}$, can someone explain why there is no permutation $a$ such that $a(1,2,3)a^{-1} = (1,3)(5,7,8)$?
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4answers
155 views

Find the number of ways to form 15 teams out of 15 men and 15 women.

In how many ways can 15 teams be formed, each consisting of a man and a woman, from 15 men and 15 women. This looks like the same problem as finding the number of bijective functions from a set $A$ ...
2
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1answer
30 views

What is the process behind finding a Cayley permutation representation.

For example, let's find the Cayley permutation representation of $\mathcal D_3$ in $S_6$. $\mathcal D_3 = \left<r,s \mid r^3=s^2=1, rs=sr^{-1}\right>$. Write, \begin{pmatrix} 1 & 2 & ...
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2answers
61 views

How do I find the permutation with the highest order in a symmetric group?

My professor gives this text, but I don't understand what it's saying, could someone explain it to me? Let $M(n)$ denote the largest order of an element in $S_n$. By Theorem 1 $M(n)$ is the largest ...
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1answer
12 views

A low-discrepancy or quasirandom series which would guarantee all value sequences

I am trying to find a type of quasi-random sequence which would guarantee that it could produce all possible sequences of values within the possible value range, while still producing random-seeming ...
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1answer
41 views

Permutations regarding $3$ women and $4$ men

$3$ women and $4$ men are standing in a line. If no two women may be adjacent to each other, how many distinct line-ups are there? I'm not sure how to do this. I know $4! = 24$ and $3! = 6$. Where ...
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2answers
60 views

Formula to calculate password cracking time in years, taking into account Moore's law and known adversary guessing power [closed]

We know that the biggest human rights violators in human history are capable of one trillion password guesses per second as of approximately January 2013. Assume that the 1 trillion guesses per ...
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1answer
43 views

Maximum number of points of intersection

The greatest number of points of intersection of 8 straight lines and 4 circles are? My attempt:Assuming every line cuts all the four circles at two points each, the points of intersection of lines ...
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1answer
27 views

Ways of forming a committee

Four couples (husband and wife) decide to form a committee of four members.Find the number of different committees that can be formed in which no couple finds a place is? My attempt:In one case where ...
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1answer
43 views

To find number of questions 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,…,n. If the total number of wrong answers given is 2047,what is the ...
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0answers
28 views

Recursive formula for the number of $n$-permutations with $k$ cycles

Let $n$ and $k$ be a positive integers satisfying $n\geq k$, then $$c(n,k)=(n-1)c(n-1,k)+c(n-1,k-1)$$ where $c(n,k)$ denotes the number of $n$-permutations with $k$ cycles. The proof of this ...
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2answers
25 views

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
45 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
56 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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2answers
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
56 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 ...
2
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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 ...