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

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How do we solve these permutation and combination questions? [on hold]

Q1 In how many ways a panel of six doctors is selected from five surgeons and six physicians if condition is surgeons are more than physicians. A 82 B 81 C 65 D 135 Q2 Find the no. of ...
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2answers
35 views

How can I simplify this number theory problem?

Let X = {1, 2, 3, 4, 5, 6} and σ= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 4 & 3 & 5 & 2 & 1 \\ \end{bmatrix} Define a relation ∼ on X as ...
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0answers
7 views

Maximize the mutual permutation disparity

I am trying to work on a problem that needs me to find the top-k most disparate permutations for a n-tuple (hence n! possible choices). The disparity measure between two permutations I'm thinking of ...
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0answers
18 views

$A^{\pi}$ property

Can someone give an example for the matrix mentioned in the following definition. Definition is taken from "Iterative Solution Methods" of Owe Axelsson. Definition: The matrix $A$ said to have ...
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0answers
17 views

How do I calculate such possible number of total and serial schedule?

Consider the following two transactions $T_1$ and $T_2:$ How many non serial schedules are possible, if we execute both transactions concurrently? $3000$ $3001$ $3002$ $3003$ My try: ...
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3answers
30 views

How to find the Permutation in S8 given as the product C= (1483)(12765)(34687)?

I don't understand how to answer this. Just by reading it off, "1 goes to 4, and 4 goes to 6, thus 1 goes to 6" but that logic doesn't match with the answer i've been giving: Answer: (127)(386)(45). ...
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2answers
29 views

3 men have 4 coats , 5 waistcoats and 6 caps. Then in how many ways can they wear them?

The question is in the title itself. First, I would like to share how I solved this problem at first: We have $4$ coats, $5$ waistcoats and $6$ caps. So, I considered that each man wears one coat, ...
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2answers
35 views

100 shoelaces, pick 2 random ends and tie them together, what is the probability that a loop is created?

The question is: There are 100 shoelaces in a box. You pick two random ends and tie them together. Either this results in a longer shoelace (if the two ends came from different pieces), or it ...
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1answer
28 views

Simply showing the addition of permutations

How can I show for example AB+BC+AC simply. It is adding up the permutations of n numbers. Another example would be ABC+ABD+ACD+BCD. Sorry I'll try to make it clear with an example ( which is sort of ...
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1answer
14 views

Number of recursive permutations of all sizes

Consider you have a set of $n$ elements. Now, create all the possible permutations of $k$ elements. Finally, for each permutation create all the possible combinations with the permutations of the ...
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1answer
41 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. ...
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2answers
58 views

number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$

Let $A=\{1,2,3,4\}\;,$ Then total number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$ $\bf{My\; Try::}$ If $f(x)=x\;,$ Then $f(f(x))=x.$ So there are ...
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1answer
14 views

Product of permutations and subgroup generated by permutation(s)

I'm get confused while working with permutations, so I have some questions. $\sigma$ = (1,7,3)(2,10,4,8) $\rho$ = (3,7) $\tau$ = (1,7) First I am told to compute $\tau$$\sigma$$\tau^{-1}$ I dont ...
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0answers
21 views

Combination and Permutation How many words can be formed? [duplicate]

A contest consists of finding all code words that can be formed from the letters in the word "alpha".Assume that the letter "a" can be used twice but the others at most once: a)How many five-letter ...
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1answer
39 views

How many 4 digit pins on set {0-9}

A password can be any 4 digit {0...9}. 1.)How many possible passwords are there? for this I did $10^4 = 10,000$ 2.) How many possible passwords with no repeated digits? $10*9*8*7 = 5040$ 3.) How ...
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1answer
25 views

5 letter password either lowercase or uppercase

Given that you can have 5 letter password that contains either lowercase or uppercase. My questions are: 1) How many possible passwords are there? I did $52^5 = 380,204,032$ since there are 52 ...
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2answers
29 views

4-Sequences {0…9}

My questions are given the set {0,1,2,3,4,5,6,7,8,9}, 1) How many 4-sequences are there? (would this be $10*10 * 10 * 10 = 10,000)? $ since the max possible numbers given to each 4 slots is 10. 2) ...
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1answer
16 views

Odds of an event happening

Trying to get my head around the correct way of approach this. You are able to use any letter of the alphabet or number allowing for 36 options, with this you are to create PIN of length 4, for ...
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0answers
32 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: ...
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0answers
49 views

Number of valid parenthesis

I have to find out the number of valid parenthesis.Parenthesis are of two type [] ,(). How many ways are there to construct a valid sequence using ...
3
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1answer
31 views

Possible ranks of a $n!\times n$ matrix with permuted rows

Let $a_1,\cdots,a_n$ be $n$ arbitrary real numbers. Form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the $n$ numbers given. Find all the possible ranks of such a matrix. ...
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1answer
35 views

Number of possible subsequences

Given 4 integers - $A,B,C,D$ such that $A \leq B \leq C \leq D$ (i.e they are in non decreasing order). Now we need to find number of possible non decreasing subsequences $(W,X,Y,Z)$ such that $1 ...
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0answers
59 views

Twisty Puzzle Solving Program

I'm writing a program to help me solve a twisty puzzle. In this case it's the face-turning octahedron. I'm representing the puzzle as a group with face twists as generators. The facelets are in a list ...
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0answers
17 views

Number of Non Decreasing Sequence. [closed]

I have to find the number of non decreasing sequence (A,B,C,D) such that ...
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3answers
42 views

Seating arrangements of 7 boys and 5 girls in a row.

In how many ways can these boys and girls be arranged in a row if between two particular boys A and B there are no boys but exactly 3 girls?
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0answers
35 views

Isomorphism of Non-Symmetric Matrix when Permutation-Set is given: A simple observation

Context: Consider, two $m \times n$ matrices $A, B$ such that there is a permutation $\kappa$ that such that such that $A^{\kappa}=B$ (Wielandt's notation), i.e. $A, B$ are isomorphic but not ...
2
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1answer
49 views

Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
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2answers
25 views

Proving that the index of a subgroup of $S_n$ that keeps a specific set invariant has a certain order

Let $n \in \mathbb{N}, n ≥ 2$, and $k \in \{1, 2, ..., n-1\}$, and let $A \subseteq \{1, 2, ..., n\}$ with $|A| = k$. Furthermore, let $G$ be a subgroup of $S_n$ that fixes $A$, i.e. for all $π \in ...
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2answers
39 views

Finding the Left and Right Cosets in $A_4$

I'm really struggling with a Group theory class and would love some help. HW Question is as follows. Consider the subgroups $H = \left<(123)\right>$ and $K=\left<(12),(34)\right>$ of ...
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1answer
26 views

Inversions of a permutation. Confused

Sorry for this basic question. In here, we have $2$ inversions of $1$ element (from the set $\lbrace 1,2,3\rbrace$): $$ 132, \\ 213, $$ and that $321$ is a $3$-element inversion permutation. Why ...
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1answer
24 views

Two Dimensional Choose or Combination Problem in Statistics

I have a two-dimensional choose problem. ...
2
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0answers
28 views

Galois group of polynomials

Let $f$ be an irreducible polynomial over a field K, and $\deg f = 4$, with roots $a,b,c,d$. Let $g$ be a cubic resolvent with roots $\alpha,\beta,\gamma$. And $\alpha=ab+cd, \beta=ac+bd, ...
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3answers
201 views

Product of permutation cycles, transpositions. Are there different conventions in the order?

From this answer I get that within each cycle you map each element to the one on the right, when taking the product of cycles the one on the right should be performed first, as a typical operator. ...
2
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1answer
30 views

Probability of selecting a jury

Does anyone know how to find the probability of selecting a jury of $12$ people ($6$ men and $6$ women) out of an initial group of $18$ people ($6$ men and $12$ women)? With my knowledge of ...
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1answer
20 views

Permutation test for equality between the distribution of $g$ population

I have a data matrix of 42000 observation and 12 variables I suppose to observe 12 samples of size $n_j$ from 12 indipendent random variables $Y_j,j=1,...,g$ I want do a permutation test for $$H_0: ...
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0answers
35 views

12 numbered pigeonholes and balls [duplicate]

This problem was inspired by this James Randi challenge. Given $12$ numbered ($1$ to $12$) pigeonholes and $12$ numbered balls (also from $1$ to $12$); what is the probability that a random ...
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1answer
18 views

Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
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0answers
24 views

Number of ways to allocate balls

We are given $N$ buckets and $X$ ways to allocate balls in each possible pair of buckets. How many ways of distributing balls in all buckets exist ? For example - If we have $3$ buckets and $7$ ways ...
0
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1answer
22 views

Character of a representation on $S_3$ and irreducible representations

Here is the character table of S3: Consider $V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ with basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ Let $\pi$ be the representation of ...
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2answers
49 views

How to calculate permutation $(12)^{-1}(12345)(12)$ [closed]

I was wondering if someone could help me find $(12)^{-1}(12345)(12)$ I need to know this for calculating conjucacy classes and then a character table, thanks
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0answers
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Algorithmic complexity of testing whether a permutation belongs to a subgroup generated by a set of permutations

Let $S=\{S_1,S_2,S_3,\ldots,S_m\}$ be a set of permutations on $n$ symbols (in other words $S$ is a subset of a symmetric group on $n$ symbols) and $P$ be a permutation on $n$ symbols. What is the ...
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0answers
22 views

What happened to the “permutation-groups” tag? [migrated]

There used to be a "permutations-groups" tag, which I don't see anymore. What happened to it? Can it be put back?
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2answers
37 views

What does this definition of permutation mean?

A simple question. They give the definition of permutation as "a one to one mapping of the set onto the set of positive integers $\{1, 2,3,4, \ldots n\}$." What does this definition exactly ...
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0answers
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K-permutation of given multiset

Is there a formula (or an algorithm) to calculate K-permutation of given multiset? The particular example: given a multiset of 15 balls in which every 3 are marked with one of the letters A, B, C, ...
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1answer
27 views

Permutations acting on coordinates of codewords

Let $\mathcal{C}$ be a binary code of length $n$. The automorphism group of $\mathcal{C}$ is defined to be the set of permutations in $S_n$ that take $\mathcal{C}$ to itself. The text by MacWilliams ...
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1answer
32 views

Permutation and Combination to find pairs

In how many different ways students can be paired such that no pair consists of 2 boys. Given :- Total students = 10, Girls = 7, boys = 3. What my approach is 3 boys can be paired with 7 girls like ...
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3answers
62 views

Number of permutations which are products of exactly two disjoint cycles. [duplicate]

Let $l_{n}$ denote the number of those permutations $f$ on the set $A=\{1,2,....,n\}$ such that $f$ is the product of exactly two disjoint cycles. Show that $l_{5}=50.$ I tried a lot but reached ...
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2answers
25 views

Understanding a proof about permutations from P.A.Grillet's “Abstract Algebra”

I need a hand in understanding the following proof of the following theorem(by P.A.Grillet in his textbook "Abstract Algebra"). Proposition $4.1$. Every permutation of $\{1,...,n \}$ is a product ...
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0answers
34 views

Find all permutation sequences with only neighbor swaps

I am trying to essentially do what was done at this link, except for five elements instead of four. The link gives all of the possible sequences of permutations of four elements with the following 2 ...
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3answers
62 views

Five people have applied for three different positions in a store. In how many ways can the positions be filled?

Five people have applied for three different positions in a store. If each person is qualified for each position, in how many ways can the positions be filled? Can someone tell me if I have to ...