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

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1answer
55 views

How many ways are there to arrange k out of n elements in a circle with repetition?

If you a set of the n elements, in how many ways $Q(n,k)$ can you take some of them and arrange them on a $k$-gon, when repetition of one element is allowed but rotations of one arrangement are not ...
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3answers
58 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
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0answers
15 views

Distance between element and its position in permutation

Consider a permutation $\pi$ that is chosen uniformly at random among all permutations of $\{1, \dotsc, n \}$. Let $a_i$ be the position of $i$. We want to find $$E[\sum_{i=1}^n |a_i -i |]= ...
2
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0answers
30 views

Need a hint with permutations and pigeonhole-principle question

let $\pi_1,\pi_2,\pi_3\in S_{28}$. Help me prove that there are two sub-sequences of 28 with length 4 $i_1< i_2 <i_3<i_4,\ and\ \ j_1<j_2<j_3<j_4$ so that $\pi_q(i_n)=\pi_p(j_n)$ ...
1
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1answer
39 views

Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
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3answers
36 views

A test contains 10 T/F questions, 5 must be marked true, and 5 false…

A quiz consists of ten true/false questions. a) In how many distinct ways can the quiz be completed if no answers are left blank? b) In how many ways can the quiz be completed if five questions must ...
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1answer
25 views

How many different vertical arrangements are there of 10 flags if…?

How many different vertical arrangements are there of 10 flags if 4 are white, 3 are blue, 2 are green and 1 is red? I know the answer is 12 600 but am not sure how to get to it. Could someone walk ...
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0answers
11 views

Calculating Permutations of multiple, non-equal shapes on 2-Dimensional grid

I originally posted this in StackOverflow before finding this Mathematics forum. This is primarily a mathematics question so I am reposting here: Situation: The following components exist for this ...
2
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0answers
55 views

Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
0
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1answer
24 views

A circle and six sectors

A circle is divided into six sectors and six numbers 1,0,1,0,0,0 are written clockwise,one in each sector. in one step, we can add one to the numbers in any two adjacent sectors. Is it possible to ...
0
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1answer
36 views

Lagrange Theorem: left and right cosets

Thank you for taking the time to read this and help me out. This was a question I missed on my past test: "Let $G = S_3$ and $H = A_3$ ($H \subseteq G$ is a subgroup). Compute the left and right ...
0
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1answer
29 views

Seven people are interviewed for a possible promotion. In how many orders can the seven candidates be interviewed?

Seven people are interviewed for a possible promotion. In how many orders can the seven candidates be interviewed? I know the answer is 5040 but don't know how to get to it.
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3answers
30 views

PIN number consists of four letters, how many different PINs are possible?

The personal identification number (PIN) used by a certain automatic teller machine (ATM) is a sequence of four letters. a) How many different PINs are possible? Write the answer in exponential ...
0
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1answer
28 views

Number of permutations of n numbers with given constraints

Given a set S of m unique numbers, n slots are to be filled using those m numbers. What are the number of ways to do it, given the following constraints: A particular number from those m numbers ...
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2answers
60 views

What does alternating mean?

My teacher ask a question to me. Question is: Determine in how many ways can be rearranged the letters of the word ECEHUCDE so that the consonants and vowels are alternating. I said it must be ...
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2answers
48 views

Combinatorics: premutations with repetition?

I have following problem from combinatorics: Let's have set of 8 distinct items: {a,b,c,d,e,f,g,h} How many ways we can combine 10 of them if we know: We start with A and end with H ...
2
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0answers
19 views

Different permutations of n identical groups with sizes a, b, and c.

My question is trying to solve how many paths there are from $(x,y)$ to $(tx, ty)$ 3 possible moves are allowed at each step: increment $x$ by $1$ increment $x$ by $2$ increment $y$ by $1$ I know ...
0
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3answers
479 views

Counting arrays with gcd 1

I want to calculate the number of arrays of size $N$, such that for each of it's element $A_i, 1 \leq A_i \leq M$ holds, and gcd of elements of array is 1. Constraints: $1 \leq A_i \leq M$ and $A_i$ ...
2
votes
2answers
75 views

Santa is secretly deranged! or, how to hand-generate assignments for a gift exchange?

Consider a standard Secret Santa/gift exchange game draw. We have a pool of $n$ people, each of whom is supposed to be assigned another member of the pool to find a gift for, without the recipient ...
3
votes
1answer
28 views

Why does in a permutation indices move opposite to positions

This is a small notational observation I first noticed when learning about permutations. I am embarassed to admit that I still do not have a satisfactory explanation for it. An example of the ...
0
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1answer
23 views

How to calculate Permutations if you have more places than distinct objects

I'm having trouble using a Permutation formula for finding out how many different ways there are to seat 264 people at 481 desks. The trouble I'm having is that in the Permutation formula (nPr = n! / ...
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2answers
32 views

Word Permutations

How many words can we build using exactly 5 A's, 5 B's and 5 C's if the first 5 letters cannot be A's, the second 5 letters cannot be B's and the third 5 letters cannot be C's? Can anyone help me? I ...
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2answers
32 views

How many decimal strings of length 55 contain exactly ten 7s?

Some more questions, I believe I have them right but I have no way to check my answers. How many decimal strings of length 55 contain exactly ten 7s? The way I think this one is answered is by ...
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4answers
41 views

How many sets of size five are there from the natural numbers 1-100 that contain exactly two odd numbers?

Hey I would like to know the correct approach to this problem. I think I'm close but I'm confusing myself. How many sets of size five are there from the natural numbers 1-100 that contain exactly ...
0
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2answers
45 views

Combination Counting puzzles me…

A test has $6$ questions with $4$ possible answers for each (a,b,c,d), plus $5$ more true or false questions. How many students are required to take the test to guarantee that $2$ write down the ...
2
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2answers
33 views

Finding a unique Mobius Transformation

Let $z_1, z_2, z_3$ be three distinct points in $\widetilde{\mathbb{C}}$. (1) show that there is a unique mobius transformation $g$ such that $g(z_1)=0, g(z_2)=1, g(z_3)=\infty$ (2) show ...
0
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2answers
47 views

Combinatorics: pick 2 from group A with 5 elements and 3 that stay in order from group B with 6 elements?

So basically: how many way are there to pick 2 elements from group A which has 5 elements, and pick 3 elements from group B which has 6 elements, provided the 3 elements stay in order in the new 5 ...
1
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1answer
26 views

Unique calendars

We have to make 3000 unique calendars. There are unique in the sense that each calendar will have twelve designs (one for each month) in such a sequence that no two calendars are exactly identical. ...
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0answers
37 views

Does each element of $D4$ have an inverse in $D4$?

We are just starting the concept of permutations of objects in my class and I'm having trouble to grasp this particular question. I'm assuming it does have an inverse because of all the different ...
0
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2answers
26 views

How many permutations of pixels in a square?

Given a square of dimensions x by y pixels, how many permutations of colors of pixels are there in the square? Assume that each square is 1 pixel and that this square is 5x5 pixels. How many unique ...
0
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1answer
31 views

Calculating the probability of a combination with repetition

If I have n colors, (Let's say n=3 , blue=0, red=1 and green=2). And we have r boxes (Let's say r=5). So we have these combinations (with repetition). ...
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1answer
75 views

Automorphism group of the Alternating Group - a proof

I was trying to read the following lemma which admit as an easy corollary the structure of the automorphism group of the alternating group on $n\geq 7$ elements. Anyway there are two points that ...
0
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1answer
39 views

permutations with the English alphabet

How many four-letter words, using the English alphabet, are possible if letters if only vowels may be repeated? How many four-letter words are there if at most one repetition of any letter is allowed? ...
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2answers
23 views

A rental car agency has 12 identical cars available and 7 identical vans…

My question is: A rental car agency has 12 identical cars available and 7 identical vans a) If the group needs to rent four cars and two vans, in how many different ways can they select their ...
0
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1answer
47 views

to find total number of subsets

I was working out some problem where I needed permutation and combination. I took the cartesian product of $n$ sets where number of elements in each set is even and $n$ is odd. Further the elements of ...
2
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1answer
97 views

Combinatorics question with stars and planets [closed]

Assume that a small universe has 10 distinct stars and 100 distinct planets so that 20 of them are habitable and 80 of them are nonhabitable by humans. How many ways are there to form a galaxy with ...
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1answer
25 views

$D_6$ and cycle notation problem

I have a hexagon with edges $A,B,C,D,E,F$ and I want to work with its symmetry group $D_6$ in cycle notation. My calculations don't yield consistent results. For example, I correctly get $r^4 \cdot ...
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2answers
18 views

Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$…

Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$,while $4$ appears in a different cycle of $\pi$ from $1,2,3$? My attempt:For $n=4$ I ...
0
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3answers
45 views

Calculating possible combinations for 1-3 digit code

I have been researching a lot on permutations and calculating total numbers of combinations of certain array lengths allowing certain characters. However, all the equations used to do this only ...
0
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1answer
20 views

One-To-One functions

Let A be the set with n elements and B be the set with m elements. How many one-to-one functions are there from A to P(B) (power set of B). There are n! total functions from A to B. and (2m)n from A ...
0
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1answer
26 views

Symmetries of a $9$ puzzle (Rubik's Slide)

Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also ...
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0answers
45 views

Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
2
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2answers
38 views

How many 2m-permutations, consisting only of cycles of even length?

How many 2m-permutations, consisting only of cycles of even length? I have found this formula: $$Q_2(n) =((2n − 1)!!)^2$$ but how it can be proven?
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1answer
37 views

fixed length of permutaions cycles

How much permutations has only 10 cycles, but three of them has length 3 and seven of them has length 7?
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1answer
14 views

Permutation test and p-value

I construct a permutation test in order to see If two samples come from the same distribution or not. I have two vectors $x, y$ that hold values of sampled values from two populations and the test ...
0
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1answer
47 views

Problem Solving Involving Permutation

Find the number of 6-digits number with no 3 consecutive number with same digits. Note that 0 might be the first number. I have tried to find the number with no pairs, 1 pairs, 2 pairs and 3 pairs. ...
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1answer
34 views

example for permutizer group

permutizer of a subgroup H of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with H. You can help us give an example?
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0answers
58 views

Multi-ruled combinatorics problem (need this for my lab)

I need to know this for practical purposes and not homework, learning etc.. Say I have 3 electrodes A,B and C. Say I also have 3 electrolytes A,B and C. If electrode A has to be in electrolyte A, ...
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2answers
41 views

let $D_n$ be the number of permutations of $\{1,2,3,…n\}$ which leave no element fixed.

Let $n\geq2$ and let $D_n$ be the number of permutations of $\{1,2,3,\dots,n\}$ which leave no element fixed. How to write an expression for $D_n$ in terms of $D_k$? I don't know how to start. Please ...
1
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1answer
100 views

Unique permutations from set with repetitions

I am new to combinatorics and might ask a trivial question: There are $69$ different items, each present $4$ times. From this total of $276$ items, $20$ should be picked at random. I need the formula ...