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

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Conjugation of permutations

In the group $S_n$ I usually use the fact that if $(a_1 a_2 \dots a_r) \in S_n$ is an r-cycle and $\sigma \in S_n$ then $\sigma (a_1 a_2 \dots a_r)\sigma^{-1} = (\sigma(a_1)\sigma(a_2) \dots ...
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estimations in the birthday paradox?

The birthday paradox is the famous following problem: What is the probability $p_n$ that at least $2$ persons amongst $n$ persons chosen at random have the same birthday? Leap years are not taken ...
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1answer
51 views

Transitive action of the group implies isomorphism with the quotient by stabilizer

Let $\Omega$ be a set and $G$ a subgroup of the group $Sym(\Omega)$ of permutations of $\Omega$. Let $\omega \in \Omega$ and let $G_{\omega}$ denote the stabilizer of $\omega$ in $G$. If $G$ acts ...
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1answer
33 views

Combinations without Repetition [closed]

For calculating follow Combinations RRDD RDRD RDDR DRRD DRDR DDRR answer is $\frac{4!}{(2!\times2!)}$. First divide into $2!$ is because does not matter order ...
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1answer
57 views

Permutations of cards with no adjacent pairs

We have a standard 52-card deck, and are looking at the possible shuffles/permutations of this deck. However, we have rubbed off the suits from the cards, so for every rank (aces, tens, etc.) all ...
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1answer
73 views

What is the use and motivation for this particular concept in permutations?

Say you have the permutation $(54231)$ element of $S_5$ Now you drop say the "4" and then re-rank the remnant permutation on the other elements. Then you are left with, $(4231)$ element of $S_4$ ...
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0answers
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Mixing time of three particle systems

Is there anything known about mixing time of Markov chains for three particle systems? It is proved here that the mixing time of an exclusion process is $\operatorname{O}(n)$. We can think if a ...
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1answer
32 views

Anagram with condition on last letter

How many ways can "computer" be arranged with a vowel as last alphabet? Isn't it $7! \times 3 $? since there are 3 vowels. $3$ (e,o,u) $ \times 7!$(number of arrangement without one of vowel). ...
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28 views

Permutation and Combinations Problem.

There are m copies of each n books on different subjects in the college library. The number of ways in which one or more books can be selected is ...?? I have no idea to deal with this problem , ...
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1answer
38 views

Natural action of $S_n$ on $\{ 1,2,\dots,n \}$

From reading online the "natural" action of $S_n$ on $\{ 1,2,\dots,n \}$ is $(g,x) \mapsto gx$. How is this action transitive? As far as I can see if we take $g$ to fix some element we will not get a ...
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2answers
111 views

How many 2 digit even numbers can be formed from these numbers?

How many even 2 digit numbers can be formed from the numbers 3,4,5,6,7? The digits cannot repeat (you can't have 44 or 66 for example). I know the answer to this is 8, because I just wrote them all ...
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2answers
28 views

More $1$s than $0$s in recursively defined set?

Let $S$ be the set of strings defined recursively by: Basis Step: $1 \in S$ Recursive Step: If $s \in S$, then $01s \in S$, $10s \in S$, $0s1 \in S$, $1s0 \in S$, $s10 \in S$, $s01 \in S$, $s1 \in ...
3
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1answer
21 views

How many permutations will there be to this problem?

How many permutations of the following pattern will there be. The order has to stay the same. In other words, you can only swap the 'B' with another 'B' because it will not affect the pattern? B C B ...
0
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1answer
62 views

Rolling 1 die 5 times [closed]

One die is rolled five times. How many different results are possible? Of those, in how many ways can there be exactly 2 rolls of 4? For the first part I multiplied 6 five times and got 7776. I'm ...
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2answers
34 views

Permutations with repetition element condition

I'm trying to figure out: How many permutations (with repetition allowed) does A,B,C have for a given $k$ (the length of the permutation) if A cannot be followed by a C anywhere in the end result? ...
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1answer
28 views

How to make 4608 combinations with these choices? (Probability, permutations/combinations)

This problem has been giving me a lot of trouble... Freeze King claims to offer 4,608 different ice cream cups. A customer can choose from 3 sizes, 4 flavors; a waffle cone, sugar cone, or cup; ...
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1answer
22 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
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1answer
31 views

How do I get number of combination for pairs of football teams?

Suppose we have 8 football teams playing each other in 4 matches. How do I find the number of combinations that is possible? E.g. Teams A,B,C,D,E,F,G,H can be in the following matches: Match 1: A vs ...
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1answer
25 views

Product of any two non disjoint cycles

Suppose you have a permutation $\sigma = \sigma_1 \sigma_2$ where $\sigma_1 = (i_1 i_2...i_k)$, $\sigma_2 = (j_1 j_2...j_l)$ and $i_{k_1},i_{k_2},...,i_{k_r}$ are equal to ...
0
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1answer
19 views

Is it possible to develop function that returns the number (rank, position) of a particular permutation.

I'm working on a data warehousing project and need to assign a unique value to a permutation and store that value as dimension in the data warehouse. Currently, I'm relying upon a rather large lookup ...
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1answer
39 views

Dividing $n$ identical things into $r$ groups

I was reading a course on Combinatorics where I came across following: The number of ways in which $n$ identical things can be divided into $r$ groups so that no group contains less than $m$ items ...
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3answers
59 views

Different ways of giving away 35 coins to 5 people?

The first part of the problem asks how many ways there are to give away 35 identical coins to 5 people, and I've concluded that it's ${35 \choose 5}$ because you're selecting how many ways you can ...
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1answer
52 views

How many different ways can you choose a group of 4 people?

You have a total of 9 people to choose from. Of these 9 people you are supposed to create a group of 4. How many different ways can the new group look? This is my reasoning: To the new group, the ...
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1answer
34 views

Generating Constrained Random Distributions

I am trying to help another StackExchange user. We are attempting to fill a 6x6 matrix with 12 A's, 12 B's, and 12 C's subject to the constraint that each row contains 2 A, 2B and 2 C and each column ...
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1answer
55 views

How to find a permutation of a specific rank?

I have a problem regarding permutations. When the rank of an unknown $S_7$ permutation is given, I want to find this permutation, but I can not. For example, I have the following questions: ...
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1answer
36 views

find the number of one-to-one function $[\pm n] \rightarrow [\pm n]$

the permutaion of $[\pm n]$ is a bijective (one-to-one) function $\pi:[\pm n] \rightarrow [\pm n]$ so that $\pi (-i) = -\pi(i)$ . $[\pm n]:=\{1, \dots, n-1, \dots, -n\}$. i have to find and determine ...
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2answers
55 views

Find a Four-element Abelian Subgroup of $S_5$ [duplicate]

Prof. Charles Pinter's "A Book of Abstract Algebra" provides this exercise: Ch 7 (Groups of Permutations) Part B #3 - Find a four-element abelian sub-group of $S_5$. Write its table. Please ...
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1answer
61 views

Which other “exotic” permutation-related things exist?

Some time back I posted some questions about the "exotic" outer automorphisms of $S_6$, and part of the answer was a citation of a paper by T. Y. Lam that said, among other things, that the ...
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4answers
102 views

How to write the set of all permutations on a set $n=\{1, 2, \ldots, n\}$

Let $n ∈ N$. Let $S_n$ denote the set of permutations on $\{1, . . . , n\}$. For any $σ ∈ S_n$, define $sign(σ) := (−1)^N$ , where $σ$ can be written as the product of $N$ transpositions. Now, let ...
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How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$?

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$ so that each column and each row has exactly $n $ $1$'s and $n$ $-1$'s ? I tried for cases $n=1 , 2$ but the solutions were ...
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1answer
44 views

Number of ways to place 4 girls into 3 bedrooms.

A family has 4 girls and 3 bedrooms. 2 of the bedrooms are only big enough 1 girl, and the last room is big enough for 2 girls. How many ways are there to assign the girls to the bedrooms? I came up ...
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2answers
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Exotic maps $S_5\to S_6$

This section says: There is a subgroup (indeed, $6$ conjugate subgroups) of $S_6$ which are abstractly isomorphic to $S_5$, At this point I'm thinking: certainly: the group of all ...
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How many possible six-word sentences

A word is defined as a nonempty (possibly meaningless) sequence of letters. How many $6$-word sentences can be made using each of the $26$ letters of the alphabet exactly once? Generalise the result ...
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1answer
72 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
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2answers
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Trying to determine the number of possible combinations for a password

OVERVIEW: Making a secure password. People tend to use dictionary words as a basis for their passwords. People tend to make minor substitutions on their passwords (password -> p@$$w0rd) Assuming ...
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1answer
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Combination or Permutation

Q.1)"Find the no. of ways in which $5$ boys and $3$ girls can be seated in a row so that no two girls are together." Q.2)"In how many ways can $5$ white balls and $3$ black balls are arranged in ...
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Characterization of conjugacy classes of $A_n$: intuition

Note the following theorem (quoted after handouts by Keith Conrad (UoCT) found online): Let $\pi \in A_n$. Its conjugacy class (cc) in $S_n$ remains the same in $A_n$, or it breaks into two cc's of ...
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2answers
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Combination and Permutations: How many ways can an award be given? [closed]

Have this Math question which I'm helping my cousin with but struggling to make sense of the answer. Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of ...
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0answers
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Idemptent of Young Tableaux

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of ...
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1answer
41 views

Permutation with repetition and restriction

There are 5 red flowers, 4 blue flowers and 4 green ones. I must plant them so that no 2 red flowers are planted near each other. So I took all the possibilities (13!) and subtracted the ones where ...
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52 views

Permutation and Combination Problem-word arrangement

There are three pieces of paper.In the three papers ,a string (non-empty) has to be written such that none of the string on any paper is prefix of some other string.Also alphabet size of characters ...
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3answers
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Number of different colourings of nodes

Consider a tree where each node has 2 subnodes, with a total of 7 nodes. So the maximum level of the tree is 2. Each node can be coloured white or black. Two colourings are equivalent if the one is ...
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1answer
39 views

A problem on permutation

Question :If there are $6$ periods in each working day of a school,in how many can one arrange $5$ subjects such that each subject is allowed at least one period? My solution: ${^5P_5} *{^5P_1}=600$ ...
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1answer
30 views

multiset/combination question

I have a bag full of: 7 green rocks, 12 yellow rocks, and 15 red rocks. How many ways are there to reach in and grab 4 rocks? Is the answer 37C34 (37=7+12+15+4-1) or 6C3 (6=3+4-1)...or something ...
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2answers
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A question about irreducible representation of symmetric group (permutation group) in tensor space and tensor contraction

In chapter 13 of the textbook of Group Theory in Physics by Wu-Ki Tung, Lemma 2 discusses the equivalence of two irreducible representations of GL(m) on ${T^i}_j$. In its proof, it simply mentioned ...
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1answer
19 views

Number of permutations of a string with consequtive repetitions disallowed

Q: How many permutations of a string $AAABBCCDD$ exist such that consequtive characters $AAA$, $BB$, $CC$ and $DD$ don't appear in it. Note that $AA$ on its own is OK. A: The total number of ways in ...
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1answer
50 views

How to do permutation questions like this one:-

I am always confused on how to answer questions like this : Find the total number of possible permutations of all the letters of the word RESERVE. Find the number of these permutations in each of the ...
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Can there be two non-isomorphic sets of permutations with a one-to-one match between i's in S1 and k's in S2 (see description)?

In thinking about this question — where sets of $M$ permutations of length $N$ ($M<N$) are defined as "isomorphic" if one permutation function can be found that, when applied to each permutation in ...
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2answers
29 views

How many ways to come up with four teammates?

A Math teacher has to choose 4 people for a competition. If there are 6 boys and 6 girls, and the teacher must select 2 boys and 2 girls, how many ways are there? I came up with ${4 \choose 2} \cdot ...
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1answer
43 views

How many ways to line up if daughters are on sides of mother?

If we have a mother, father, 2 daughters and 3 sons lining up for a family photo, and the mother must be between the daughters, how many ways are there for the family to line up? I came up with ${5 ...