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

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2answers
39 views

What are all the elements of the group of symmetries of a regular tetrahedron?

I can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on 'top' and we do to the other $3$ as we did in a ...
0
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2answers
26 views

How many ways are there to choose 2 numbers such that their product is a multiple of 3

Let A be the set A = {1; 2; 3; ... ; 20} containing natural numbers from 1 to 20. How many ways are there to choose 2 numbers for A such that their product is a multiple of 3? I tried to take the ...
2
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1answer
16 views

size of group of row and column flips of a square board

Let $X$ be the set of numberings of the squares in a $n \times n$ board with the numbers $1$ to $n^2$. Let $G$ be the group of transformations of boards generated by row and column flips, where a flip ...
1
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1answer
37 views

How many combinations of permutation matrices are there?

I want to know, for an $n\times n$ permutation matrix, how many matrices are there such that there are exactly $3$ entries above the diagonal. For example, there is only one $4\times 4$ matrix that ...
0
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1answer
29 views

Arranging m distinct groups of k elements each?

There are n elements in total which is already divided into m groups of k elements each. Thus, n=m*k. The question is, how many arrangements of these m groups are possible? I came to a possible ...
0
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1answer
38 views

Combinatorics - Selecting Finite Possible Pairs from Infinite Pool

A friend of mine asked me this puzzle few days back. I have been trying to solve this. I tried it by doing manually creating possible pairs of yellow & blue ribbons But I am sure there has to be a ...
0
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1answer
43 views

find normal subgroup of symmetric group

Symmetric group $S_3=\{(),(1,2),(2,3),(1,3),(1,2,3),(1,3,2)\}$, I understand that $H=\{e,(1,2,3),(1,3,2)\}$ is the normal subgroup of S3 ($H\lhd S_3$) because: $$ gH=Hg, \forall g\in S_3$$ e.g. let ...
1
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1answer
26 views

Prove that every permutation in $S_k$ is the product of transpositions of the form $(j, j + 1).$

Prove that every permutation in $S_k$ is the product of transpositions of the form $(j, j + 1).$ I proved the case $n=2$ for my base case... so $(12)=(21)$ and $(21)=(12)(12)$ then I proved $n=3$ and ...
0
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3answers
42 views

The number of words containing four $a$'s and two $b$'s

Find the number of words containing four $a$'s and two $b$'s. I thought of $6!$ but then I found out that there will be many repetitions in that case. Thanks in advance.
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1answer
22 views

Parallel lines with points [closed]

There are 2 parallel lines. One of them has 5 points on itself, and the other one has 4 points on itself. How many triangles are there whose vertices are those points. Thanks in advance.
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1answer
32 views

How many ways are there to choose $6$ children from $7$ boys and $4$ girls on condition that at least one is a girl? [closed]

There are 7 boys and 4 girls in the kindergarten. How many ways are there to choose 6 of them on condition that at least one is a girl. Thanks in advance.
0
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1answer
21 views

How to find all stabilizater of element of the same orbit.

Let $\mathfrak S_4\times \mathfrak S_4\longrightarrow \mathfrak S_4$ the action by conjugaison, i.e. $$\sigma \cdot \tau=\sigma \tau\sigma ^{-1}.$$ I have shown that the orbits are $\mathcal ...
1
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1answer
27 views

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 ...
0
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1answer
27 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 ...
0
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0answers
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 ~~|~~ ...
2
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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 ...
0
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0answers
36 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, ...
2
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1answer
32 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 ($2^{\aleph_0}$ ...
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 ...
1
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1answer
72 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.
0
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1answer
44 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 ...
2
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1answer
29 views

string and its permutations

I have a string lets say abcd so its all permutations would be ...
0
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0answers
14 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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0answers
20 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 ...
1
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5answers
67 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 ...
1
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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 $2^{21}$, ...
1
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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 $(1,\ldots,i,\ldots,j,\ldots,m)$ ...
0
votes
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 ...
17
votes
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 ...
2
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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 ...
1
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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 ...
1
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1answer
37 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 ...
1
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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?
0
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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 ...
1
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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 ...
0
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1answer
33 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, ...
1
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2answers
54 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 ...
2
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2answers
32 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' ...
0
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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? ...
1
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1answer
38 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 ...
0
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1answer
29 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 ...
2
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1answer
14 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 ...
0
votes
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, ...
1
vote
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 ...
1
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2answers
89 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 ...
7
votes
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 ...
3
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2answers
38 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 ...
-1
votes
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? ...
2
votes
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 ...