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

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3
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1answer
79 views

What is the probability of no pair's names being adjacent?

Suppose there are $N$ (even number, positive) people. And each one person has to find one and only one partner to form a pair. There is also a roster within which everyone's name appearing in ...
0
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0answers
40 views

How many ways can two cards be selected from five different cards with replacement when order matters?

So I have $5$ different cards and I am choosing $2$. I want to know the total possible permutations of choosing two cards (I know the answer is $25$ by using permutations). I need other ways to ...
1
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0answers
27 views

Find all the invariant invariant subspaces for the regular representation of $S_3$.

Using a little program I can build the regular representation of $S_3$ (the permutation group of three elements). The textbook that I am reading only tells that the regular representation contains all ...
0
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1answer
28 views

a[i] denote number of friends i-th student has. c[j] denote frequency having at least j friends. Show that: ∑a[i]=∑c[j]. [duplicate]

Q. A class has 100 students. Let a[i], 1≤i≤100, denote the number of friends the i-th student has in the class. For each 0≤j≤99, let c[j] denote the number of students having at least j friends. Show ...
1
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1answer
30 views

Calculating probability of obtaining exactly two $20$'s in $40$ rolls of a fair $20$-sided die

I have a question: On a fair 20-sided die, the number $20$ comes up once every $20$ rolls. In forty rolls, it's expected that about two rolls of $20$ will happen. What are the actual odds that, ...
1
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2answers
101 views

Showing that a quotient group $G/N$ is isomorphic to $\mathbb{Z}_3$

I have permutations $\sigma=(135)(27)$, and $\tau = (27)(468)$. $G =\langle \sigma,\tau \rangle$ and $N$ is the smallest subgroup of $G$ that contains $\tau$, so $N = \langle \tau \rangle$. $|\sigma| =...
3
votes
1answer
24 views

Expressing a permuation as a product of disjoint cycles.

The theorem: Let $p$ be a permutation of $\{1,\ldots,n\}$. Then $p$ can be expressed as a product of disjoint cycles. How would you express a permutation that permutes every element of the set, ...
0
votes
1answer
41 views

Number of ways to arrange the alphabet, A is not in first position, B is not in second, and so on.

My first answer was 25! as The first letter has 25 possibilities, second letter 24 possibilities, and so on. However, I realised that it is actually more than that. There is a possibility that the ...
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0answers
46 views

Sum of some number of prime numbers is n. What is the upper bound on their product?

Sum of some number of prime numbers is n. What is the upper bound on their LCM (Or you can assume that all primes making up sum n are distinct)? I got some bound esqrt(n * ln(n)). Is there some ...
0
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2answers
46 views

$S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. $...
0
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3answers
39 views

How many numbers of $7$ digits can be formed with the digit $0,1,1,5,6,6,6$.

How many numbers of $7$ digits can be formed with the digit $0,1,1,5,6,6,6$. My attempt: Seventh place, total number of possibility is $=\frac{6!}{2!\times 3!}=60$ ways. Sixth place, total ...
1
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2answers
48 views

In how many ways can $5$ students and $3$ teacher sit around a table so that no two teachers are together?

In how many ways can $5$ students and $3$ teacher sit around a table so that no two teachers are together? My attempt: $5$ student can sit $(5-1)!$ in round table. A teacher can sit between two ...
0
votes
1answer
16 views

Demonstrate that the product of the permutations(regardless of order) of $S_4$ is not equal to $a$

$S_4$ is the set of all permutations of length 4 Let $a=\binom{1\,2\,3\,4}{3\,2\,1\,4}$ I found that $a$ is an odd permutation and I want to demonstrate that the product of the permutations is even ...
2
votes
1answer
29 views

How many $3$ digit different number that will be divisible by $5$ can be formed from the digit $0,2,3,4,5,6$ lying between $100$ and $1000$.

How many $3$ digit different number that will be divisible by $5$ can be formed from the digit $0,2,3,4,5,6$ lying between $100$ and $1000$. My attempt: Divisible by $5$ is possible only when ...
0
votes
0answers
33 views

Constraint for matrix representation for general irreducible permutation group.

Say I have a matrix $\bf P$ for which is ensured that $P_{ij} \in \{0,1\}$. Then consider this requirement: $$\sum_{k=0}^{n-1}{\bf P}^k[1,0,\dots,0]^T = [1,1,\dots,1]$$ Should this be enough to make ...
5
votes
3answers
294 views

How many 6 digit numbers are possible with no digit appearing more than thrice?

How many 6 digit numbers are possible with at most three digits repeated? My attempt: The possibilities are: A)(3,2,1) One set of three repeated digit, another set of two repeated digit and ...
1
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2answers
60 views

Problems on Derangements - Combinatorics [closed]

Determine the number of permutations of $ 1,2, \ldots ,8$ in which no even integer is in its natural position. Please solve this using concept of derangements.
3
votes
1answer
32 views

the MISSISSIPPI problem - 5 letter word and 6 letter word with constraints

1) Number of ways of selecting 5 letters such that 3 are alike of one type and 2 are alike of another type. There are only 2 ways of selecting 3 letters of the same type i.e., III or SSS. Now with ...
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0answers
38 views

In how many ways consonants and vowels alternatively for letters of word `CONSTITUTION`. [closed]

In how many ways consonants and vowels alternatively for letters of word CONSTITUTION. My attempt: The word 'CONSTITUTION' has 7 consonants (C N S T T T N) ...
0
votes
1answer
28 views

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

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 ...
2
votes
3answers
32 views

Permutation and order

I know that the order of a group is the number of the elements, then if we have a permutation what does the order of permutation mean? The number of distinct elements?
1
vote
2answers
42 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 ...
1
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0answers
12 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 ...
0
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0answers
21 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 ...
0
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0answers
20 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: ...
12
votes
1answer
233 views

Doing a magic trick with limited memory (from a problem solving course)

I got the following question in a problem solving course: There are four different objects lying on places 1, 2, 3, 4. A magician closes his eyes and someone from the audience comes. He switches ...
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3answers
39 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). ...
0
votes
2answers
32 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, ...
0
votes
2answers
52 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 ...
-2
votes
1answer
32 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 ...
1
vote
1answer
19 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 ...
0
votes
1answer
47 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. Since ...
1
vote
2answers
60 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 $...
0
votes
1answer
16 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 ...
0
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0answers
22 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 ...
4
votes
1answer
40 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 ...
1
vote
1answer
38 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 ...
0
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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) ...
0
votes
1answer
18 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 ...
0
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0answers
35 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: $\varepsilon_{ijk}\varepsilon_{ipq}\...
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0answers
55 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
votes
1answer
35 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. ...
0
votes
1answer
36 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
63 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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3answers
43 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?
3
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0answers
40 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
votes
1answer
51 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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vote
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 $π \in G$,...
0
votes
2answers
42 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 ...
0
votes
1answer
27 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 $...