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

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Permutation and induction

Each permutation in $A_k$ can be written as a product of 3-cycles of the form (1, 2, 3), (1, 2, 4),...,(1, 2, k). I am trying to start this problem by induction but I am having trouble with the base ...
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1answer
29 views

Writing permutation as a product of transpositions

I have a problem writing permutations as a product of disjoint cycles. For example, in the book, there are the following cycles: $(132)=(13)(12)$, $(1243)(243)=(23)(34)(14)$ Can someone please ...
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28 views

Counting powers of permutations

I didn't find similar questions so decided to ask this one. Given positive integers $n$ and $d$ how can we efficiently estimate (or better calculate) cardinality of the set $~~ \{ \sigma^d ~~|~~ ...
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1answer
31 views

Combinations of letters with restrictions

Create a string of five letters using the letters: A, B, C, D, E, F, G, H, I, J, K, L, M. a) How many words contain at least one ...
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37 views

Poker Probability. Multiple Questions.

this is for my own personal use, not school related. Texas Holdem. Each player is dealt 2 cards from a deck of 52 cards. Once the cards are shuffled the dealer gives each of the 9 players 2 cards, ...
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1answer
35 views

Interesting sequence of all the natural numbers [closed]

What are some sequences that contain all of the natural numbers that come up naturally in mathematics? (Obviously, there are an infinite number of sequences of all the natural numbers ($2^{\aleph_0}$ ...
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0answers
12 views

Verify $T^*(f^\sigma)$ = $(T^*f)^\sigma$

Where $T^*$ is linear. $f^\sigma(v_1,...,v_k)$ = $f(v_{\sigma(1)},...,v_{\sigma(k)}) $T^*f(v_1,...,v_k)$ = $f(T(v_1),...,T(v_k))$ Attempt at the proof: I didn't use the fact that T is linear ...
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1answer
75 views

arrangement of $n$ oranges and $n$ apples around a circle

what is the total number of distinct arrangements of $n$ oranges and $n$ apples around a round table? I have no idea how to go about.
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1answer
47 views

how many words can be formed from a scrabble rack with 7 letters

Given a scrabble rack with 7 unique letters, how many words (meaning not important) can be formed with 1 to 7 letters? My first thought was to take all the permutations from p(1,7) to p(7,7) and add ...
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1answer
30 views

string and its permutations

I have a string lets say abcd so its all permutations would be ...
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0answers
15 views

Write $\sigma$ as a composite of elementary permutations

Let $\sigma \in S_5 $ be the permutation s.t: $(\sigma(1),\sigma(2),\sigma(3),\sigma(4),\sigma(5))$ = (3,1,4,5,2) Write $\sigma$ as a composite of elementary permutations. Definition of elementary ...
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21 views

How do I show that the two methods of permutation test are both the same?

My main objective is to show the methods described below are really the same. However, I am having difficult both formulating the idea clearly and proving my assertion. Below is my attempt. Suppose ...
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5answers
69 views

Number of ways of making a die using the digits $1,2,3,4,5,6$

Find the number of ways of making a die using the digits $1,2,3,4,5,6$. I know that $6!$ is not the correct answer because some arrangements can be obtained just by rotation of the dice. So there ...
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1answer
35 views

Number of complete directed graphs with equal in- and out-degrees

What is the number of complete directed graphs with $7$ labelled vertices such that every vertex has an in- and out-degree equal to $3$? The total number of directed graphs possible is $2^{21}$, ...
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0answers
36 views

Counting the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$, where $i < j$ and number of inversions is $k$.

How can I prove the following: $d^{ij}(m,k) > d^{ji}(m,k)$ for all $k < \frac{1}{2}\binom{m}{2},$ where $d^{ij}(m,k)$ denotes the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$ ...
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0answers
18 views

Number of ordered pairs of given string [duplicate]

Given a string $A$. Definition: Two strings are said to be equal if after applying following operation, they become equal(equivalent). Operation: Given two strings, you can swap any characters in ...
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8answers
3k views

In how many of the integer numbers between 0 and 10,000 does the digit 3 appear to the left of 4

In how many of the integer numbers between $0$ and $10\,000$ does the digit $3$ appear some place to the left of the digit $4$? This would include, for example, the numbers $34$, $374$, $4384$ and ...
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1answer
32 views

Number of words such that only two $A's$ should be together

Find number of words that can formed using letters of the word "$PARABOLA$" such that only two $A's$ should be together. I did using two methods: Method $1.$ Let $AA=X$ $*P*R*B*O*L*$ now we have 6 ...
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1answer
27 views

Why can we simply pool the realized observations in a permutation test?

Let a vector of i.i.d random varibles $(X_1,X_2,X_3,\cdots, X_m)$ and another vector of i.i.d rvs $(Y_1,Y_2,Y_3,\cdots, Y_n)$ be given. Suppose $X_i$ stands for the recovery time using a new treatment ...
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1answer
42 views

permutations around a round table with labelled seats

Two men, Adam and Charles, and two women, Beth and Diana, sit at a table where there are seven places for them to sit down. Two people are sitting next to each other if they occupy consecutive ...
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1answer
27 views

How to prove the group $S_4$ of permutations (or bijections) has no elements of order 12?

I know there are no elements of $S_4$ with order 12 from a list of the elements of $S_4$ but how can I prove it without listing all the elements with their orders?
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1answer
12 views

How many words, with or without meaning, can be formed by selecting $3$ consonants and $2$ vowels from $7$ consonants and $4$ vowels?

There are $7$ consonants and $4$ vowels. How many words, with or without meaning, can be formed by selecting $3$ consonants and $2$ vowels? Should one consider permutation or combination?
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1answer
43 views

How to find number of strings generated by permuting the given string and satisfying the given constraints??

The question goes like this- How many strings can be generated by permuting the characters of the string "aaaa[Na times]...bb[Nb times]...cc....c....kkk[Nk times]" such that ...
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0answers
36 views

How to calculate the number of strings generated by permuting the given string satisfying the below conditions?

The question goes like this- How many strings can be generated by permuting the characters of "abbbbcccdeff" such that there are only 1 mismatchings and the rest 11 are same ? My attempt- In my ...
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1answer
34 views

How many ways are there to make $35¢$ change with these specific coins?

How many ways are there to make $35¢$ change with:a) $1952$ pennies, $1959$ pennies, and $1964$ nickels? (Numbers refer to the years of the coins, not the quantity.)b) $1952$ pennies, $1959$ pennies, ...
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2answers
55 views

How many positive integer solutions are there to $x_1+x_2+x_3+x_4<100?$

How many positive integer solutions are there to $x_1+x_2+x_3+x_4<100?$ I know how to approach the problem if it were How many positive integer solutions are there to $x_1+x_2+x_3+x_4=100$, it ...
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2answers
35 views

How many ways can $2$ a's, $2$ b's and $8$ c's be arranged so that there is a c on both sides of each a and b?

How many ways can 2 a's, 2 b's and 8 c's be arranged so that there is a c on both sides of each a and b? I'm really unsure of how to even begin tackling this. I would say treat each 'cac' and 'cbc' ...
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1answer
27 views

Probability of Bridge Hands Using Distributions

In a bridge deal, what is the probability that: a) West has five spades, two hearts, three diamonds, and three clubs? b) North and South have five spades, West has two spades, and East has one spade? ...
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1answer
39 views

Two men, Adam and Charles, and two women, Beth and Diana, sit at a table where there are seven places for them to sit down

Question Two men, Adam and Charles, and two women, Beth and Diana, sit at a table where there are seven places for them to sit down. Two people are sitting next to each other if they occupy ...
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1answer
43 views

Circular permutation - Arranging 4 persons around a circular table where 8 seats are there. (cond.)

Suppose 4 persons A,B,C and D sit around a round table with 8 seats. Rotation by 8,16,24,... seats defines same arrangement and other rotations gives different arrangements. If seats are ...
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1answer
29 views

Circular permutation - Arranging 4 persons around a circular table where 8 seats are there.

Suppose 4 persons $A,B,C$ and $D$ sit around a round table with 8 seats. Rotation by 8,16,24,... seats defines same arrangement and other rotations gives different arrangements. Find the number of ...
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1answer
62 views

How to find number of strings generated by permuting the given string satisfying the below conditions?

The question goes like this- How many strings can be generated by permuting the characters of "abbbbcccdeff" such that there are only 3 mismatchings and the rest 9 are same ? My attempt- Obviously, ...
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1answer
45 views

Isomorphic subgroups of finite groups

Which is the smallest number $n$ such that $S_n$ has non-isomorphic subgroups of the same order with the same number of cyclic subgroups of the same order? Example: $S_4$ has subgroups of ...
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2answers
90 views

How to find number of ways of permuting a string satisfying the below conditions?

I am given a string,let say- "abcd". Now I have to find all the strings that can be generated by permuting its character such that- There are exactly four mismatches in the generated strings ...
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1answer
77 views

Maximal twist in braid group product

Suppose I have the $s_i$ and $s^{-1}_i$ as generators, satisfying the braid relations. I call the $s_i$ "right twists" and their inverses "left twists". Any element $w$ in the braid group can be ...
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2answers
42 views

How to show that $y=Px$ is distributed like binary $x$ for random permutation $P$?

Drawing a random binary vector $X\in\{0,1\}^n$ from the uniform distribution, the probability $\mathbb{P}(X=x)$ to get a specific $x\in\{0,1\}^n$ is known ($=\frac{1}{2^n}$). Let ...
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1answer
21 views

simple combinations problem

We have a database of 100 images. We need to pick a set of 20 images. Order does not matter. It's just a set of 20 different images. No repetitions. How many possible sets of 20 images can there be? ...
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1answer
64 views

A Teacher wrote either of words $PARALLELOGRAM$ or $PARALLELOPIPED$

A Teacher wrote either of words $PARALLELOGRAM$ or $PARALLELOPIPED$ on board but due to malfunction of marker words are not properly written and only two consecutive letters $RA$ are visible, then the ...
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1answer
22 views

Summation underlying symmetric group

I know this summation has two term.How can we write ? Asuume that $\sigma(1)=1, \sigma(2)=2$ ,$$\sum\limits_{\sigma \in S_{2}} \prod\limits_{i=1}^2 d_{\sigma(i)}(x_{i})=\sum\limits_{\sigma \in S_{2}} ...
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0answers
19 views

Automorphism on $S_n$ sends given k-transposition to a transposition

For given integer $k>1$ and k-transposition $g \in S_n$, is there a isomorphism from symmetric group $S_n$ to itself such than sends $g$ to a transposition (a 2-cycle)? It seems that for $k=2$ it ...
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0answers
22 views

Number of arrangements of $n$ different objects taken $r$ at a time

Prove that total number of arrangements of $n$ different objects taken $r$ at a time when each object can be repeated $r$ times is $n^r$. Could someone please give me some hint as how to initiate ...
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3answers
54 views

Two bags are to be put with $5$ red and $7$ white balls. How must one divide the balls so we get the least chance of drawing a red ball?

The complete question is Two bags are to be put altogether with $5$ red and $7$ white balls, neither bags being empty. How must one divide the balls as to give a person who draws one ball from either ...
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3answers
91 views

How many odd numbers less than $600$ can be made from the digits $2,3,3,5,6,7$ with each only being used once?

How many odd numbers less than $600$ can be made from the digits $2,3,3,5,6,7$ with each only being used once? I've tried this multiple times but it's really confusing. There isn't a specific ...
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0answers
25 views

Find the orbit and stabilazer of the action $\mathcal D_8\times \{1,2,3,4\}\longrightarrow \{1,2,3,4\}$.

The action is $\mathcal D_8\times \{1,2,3,4\}\longrightarrow \{1,2,3,4\}$ defined by $\sigma \cdot x=\sigma (x)$ where $\mathcal D_8$ is the diedral group of order $8$. I identify $\mathcal D_8$ to ...
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2answers
35 views

All possibile combinations for certain number of boxes and numbers

I need to find out the formula to calculate the number of all possible combinations for the following scenario: There are 5 boxes. And there are numbers from 0 to 9. For each combination, 3 boxes ...
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0answers
32 views

Monoidal and classical definition of algebraic operads- equivariance

I am studying Algebraic Operads, following the book Algebraic Operads by Jean-Louis Loday and Bruno Vallette. In this book they provide many equivalents definitions for algebraic operads, and I am ...
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45 views

Algebraic operads and block permutations

I am studying Algebraic Operads, my reference is Algebraic Operads by Jean-Louis Loday and Bruno Vallette. There, they use a particular type of permutation, named block permutation. Unfortunately I ...
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3answers
103 views

Number of permutations of the word “PERMUTATION”

In how many ways we can arrange the letters of the word "PERMUTATION" such that no two vowels occur together and no two T's occur together. I first arranged consonants including one T as below: ...
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1answer
51 views

How many ways to select $r$ items from $n$ items, without selecting any two consecutive items? [duplicate]

Suppose that there are $n$ terms which are numbered from $1$ to $n$. We have to select any $r$ terms out of $n$. But there shouldn't be any two consecutive terms. In how many ways this can be done? ...
2
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1answer
23 views

Number of ways of selecting unordered pair of sets $A$ and set $B$ such that $A\cup B\subset X$

$X={1,2,3,....,2017} $ and $A\subset X; B\subset X; A\cup B\subset X$ Then number of ways of selecting unordered pair of sets $A$ and set $B$ such that $A\cup B\subset X$ will be? Answer is ...