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

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2
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2answers
125 views

$(123)$ and $(132)$ are not in the same conjugacy class in $A_4$

Could you tell me how to show that $(123)$ and $(132)$ are not in the same conjugacy class in $A_4$? I know that all 3-cycles can't be in the same class, because the order of each class must divide ...
1
vote
1answer
43 views

Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity when $G$ is infinite?

Let $G$ be an infinite transitive permutation group acting on a set $\Omega$. Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity for $G$ in $\Omega$? $G_\alpha$ is the set of elements of ...
-1
votes
1answer
84 views

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition? Here $S_9$ denotes the group of all permutations (i.e. bijections with itself) of the ...
-1
votes
3answers
151 views

Is this possible to convert one array to another given array?

you are given two arrays having n elements , like for n=4,suppose array1={1,2,3,4} array2={2,1,4,5} convert array 1 to array2 performing operation minimum number of time . Also state if ...
3
votes
0answers
66 views

Dinner group rotation. Sixteen couples. Four couples per house. Each couple to meet all the others, no repetition.

I want to set up a rotation of sixteen couples with four couples per house so that all couples eventually have dinner together, no repetition. Each couple is to host one dinner. Meetings are monthly ...
2
votes
1answer
2k views

5 Letter Arrangements of the word 'Statistics'

How many different 5-letter 'words' can be formed from the word 'statistics'? I really am pretty stumped. I understand how to calculate more simpler questions in which each letter of the word is ...
2
votes
0answers
246 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
0
votes
3answers
536 views

In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together?

Problem : In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together? 1st Approach : L T L T L T L T L The 5 lions should be arranged in the ...
1
vote
1answer
101 views

The number of permutations in a finite set such that $\sigma(i) \not= i $

Given a finite set $S$ with $|S|= n$, what is the number of permutations $\sigma$ such that $$\sigma(i) \not= i, \forall i \in S $$ That is, permutations where every element is interchanged with ...
1
vote
1answer
93 views

Orders of Cycles

Suppose $\tau$ is a cycle of order $n$. I am trying to show that $\tau^k$ is a cycle if and only if $\gcd(n,k)=1$. $\Rightarrow$ If $\gcd(n,k)=1$, then the order of $\tau^k$ is $n/\gcd(n,k)=n/1=n$. ...
0
votes
1answer
82 views

Combinations and their sum with constraints

I have a number of books (n). They all have different a different thickness and mass. I know that there are (2^n)-1 combinations to place the books. The order of the books does not matter. However ...
2
votes
3answers
857 views

How many 90 ball bingo cards are there?

In the UK there are 90 bingo balls. A bingo card consists of 9 columns and 3 rows. A row contains exactly five numbers and four blanks. A column consists of one, two or three numbers and never three ...
2
votes
2answers
3k views

A fair die is rolled eight times. What is the probability of getting exactly 2 threes, exactly 3 ones, and exactly 2 sixes?

I was given the following hint: Break the task of counting the desirable outcomes into three subtasks: (a) choose the location of the extra non-three, one, or six in the sequence of eight throws, ...
2
votes
2answers
163 views

Size of alternating group $A_n$

This is not too obvious to me - what is the size of alternating group? Following the hint in the comment, should it be $A_n = S_n/2$? So I don't feel right up to here.....
0
votes
1answer
174 views

Solving a statistics problem with both permutation and combination

The question I'm having issues with is 17, as shown in the picture. I understand the difference between combinations and permutations, though I'm having trouble applying it to the question. This ...
0
votes
2answers
618 views

all possible sequences of positive integers that sum upto N and are strictly increasing

I have $N$ bricks and i have to build a staircase. A staircase will consist of steps of different sizes in decreasing order, no two step size should be same. Each step should consists of atleast one ...
1
vote
0answers
64 views

Count possible decodings for given number

If A = 1, B = 2, C = 3,....,Z = 26 How to count possible decoding for given any integer number? EXAMPLE : NUMBER : 111 --> ANSWER : 3 EXPLANATION : ...
1
vote
2answers
151 views

Possible ranks of a matrix

Let $v=(a_1,\cdots,a_n)$ be a real row vector. We may form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the entiers of $v$ in all possible ways. The rows can be listed in an ...
4
votes
2answers
365 views

Derangement of n elements

What are the total number of ways to arrange $N$ objects such that first $K$ are deranged? I know the general formula of Derangement of $N$ objects. Is there any way the above problem be reduced to ...
0
votes
0answers
109 views

Finding elements of a symmetric group?

I am studying permutation groups. I understand that a symmetric group $S_n$ is the set of all permutations on $n$ symbols. So, for $S_4$ there will be 24 elements. i can write this like different ...
2
votes
1answer
38 views

Complexity class of generating all permutations

What is the complexity of the following problem (i.e. to what complexity class does it belong)? Given a positive integer $n$, provide all permutations of the sequence $\{1, 2, \ldots, n\}$.
2
votes
1answer
43 views

How many increasing 3 term geometric progressions can be obtained from the sequence $1, 2, 2^2,2^3,… …, 2^n$?

For example, an increasing 3-term geometric progression for $n ≤ 8$ is $\{2^2, 2^5, 2^8\}$.
1
vote
0answers
71 views

N people sit at a round table, starting from #1, every other one leaves, who's the last one?

For example, there are 10 people sitting there. So the 1st round, such people leave: $$\#1, \#3, \#5, \#7, \#9$$ and remains $$\#2, \#4, \#6, \#8, \#10$$ Then the 2nd round, such people leave: ...
1
vote
1answer
260 views

Theorem regarding greatest common divisor of certain Binomial coefficients.

Recently my friend asked following question- find the greatest common divisor of all binomial coefficient for a given n so the problem is in mathematical form ...
1
vote
1answer
692 views

How to find different number of distinct integers from given set of number

How many different integers can be expressed as the sum of $3$ distinct numbers from the set $\{3, 10, 17, 24, 31, 38, 45, 52\}$? Could someone help me with this problem?
2
votes
3answers
606 views

Combination problem distributing

In how many ways 8 candies can be distributed be among 3 children? I have done by using combination method $8C4$ but the answer is given 45. someone has done in this way: 008 in 3 ways ...
1
vote
2answers
127 views

Calculations with permutations: show that $(1,2,3)^2(5,7)^2=(1,3,2)$

How can I show $(1,2,3)^2(5,7)^2=(1,3,2)$? And, specifically, what does $(1,2,3)^2$ and $(5,7)^2$ equal individually?
1
vote
0answers
68 views

Asymmetric block ciphers?

Any block cipher transforms a block of $N$ bits into another block of $N$ bits based on a $\mathcal{K}$ bit key. This can be considered to be a substitution cipher on an alphabet consisting of $2^N$ ...
2
votes
3answers
114 views

Direct proof that Pr[2 immediately follows 1] in a random permutation is 1/n

The probability that $1$ is a fixed point of a random permutation of $\{1,2,\ldots,n\}$ (with uniform distribution) is $1/n.$ This is easy to prove since there are $(n-1)!$ permutations that have $1$ ...
3
votes
2answers
707 views

Logic for decomposing permutation into transpositions

I know that any permutation cycle can be decomposed into transpositions as follows: $(a_1,a_2...,a_n) = (a_1,a_{n-1})...(a_1,a_2)$ But in my book there is an example of the following form ...
0
votes
1answer
98 views

How many 7 letter passwords can I make using letter A,B,C?

I did it using the multichoose formula but it does not work where Order Matters. So I am stuck please help!! EDIT: You have to use all three letters.
3
votes
0answers
87 views

Counting number of distinct systems

This is an enumeration problem in conjonction with some lottery problems. Given an integer $N \ge 5$. Let a ticket be a set of 5 distinct integers between $1$ and $N$. Given an integer $T$ between ...
0
votes
1answer
137 views

Permutation group

let $G$ be a primitive permutation group of degree prime order $ p$, such that for stabilizers $ G_\alpha$ and $ G_\beta$ in $G$, $ G_\alpha\cap G_\beta=\{1\}$, prove $G$ is not simple.
1
vote
2answers
117 views

Approximation for the number of involutions?

I am interested any approximation that may be available for the following expression: $$ {\left(2n\right)}!\sum_{k=0}^{n}\frac {1} {2^k \; k! \; \left(2n-2k\right)!} $$ ... which can be expressed ...
4
votes
2answers
72 views

Proving inverses with permutations?

Prove (if f and g are permutations) that $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$. My teacher gave me the hint that it has something to do with identity mapping, but that doesn't help me at all. ...
5
votes
1answer
230 views

Is any permutation the product of two involutions?

The abstract of this paper says: "It is well-known that any permutation can be written as a product of two involutions." I was looking for any web resource that can provide an affirmation and ...
1
vote
1answer
59 views

How to find this formula in this dihedral group of transformations of the plane?

In the group of all the bijections of the Euclidean plane onto itself, let $f(x,y) \colon = (-x,y)$ and $g(x,y) \colon = (-y,x)$ for all points $(x,y)$ in the plane. Let $$G:= \{f^i g^j | i=0,1; \ ...
-4
votes
1answer
59 views

How many different variations of iPhone 5c are in the US market?

There are 5 colors variations for the iPhone 5c, each has 2 different memory sizes. Each of color/memory combination is sold in additional 5 variants: three simlocked (AT&T, Sprint and Verizon) ...
1
vote
3answers
158 views

Which elements could possibly commute with a cycle of full length in $S_n$?

In the symmetric group of degree $n$, which elements could possibly commute with the permutation $\sigma$ given by $\sigma(i) = i+1$ if $i < n$; $\sigma(n) = 1$? Of course, the permutations $e= ...
0
votes
1answer
85 views

Is the identity a permutation that commutes with all bijections of a set with more than two elements?

If $|S| > 2$, then what can we say about the bijection $f_0$ of $S$ onto itself such that $f_0$ commutes with every bijection of $S$ onto itself? Of course, the identity commutes with every map; ...
2
votes
2answers
290 views

Difference of number of cycles of even and odd permutations

Show that the difference of the number of cycles of even and odd permutations is $(-1)^n (n-2)!$, using a bijective mapping (combinatorial proof). Suppose to convert a permutation from odd to even we ...
1
vote
3answers
53 views

What will $_n\!C_r$ and $_n\!P_r$ be when $r=n$ or $r=0$?

Suppose I have $n$ items to choose from, and I take all of them, i.e., $r=n$. Then what will be the values of $_n\!C_r$ and $_n\!P_r$? I think $_n\!C_r$ will be $1$ because there can be only one ...
1
vote
0answers
870 views

what is maximum number of points of intersection between the diagonals of a convex octgon?

What is the maximum number of points of intersection between the diagonals of a convex octagon (8-vertex planar polygon)? Note that a polygon is said to be convex if the line segment joining any two ...
0
votes
3answers
254 views

Probability in Dice

What is the meaning of unbiased diced? Two unbiased dice are thrown. Find the probability that neither doublet nor a total of 10 will appear. to solve this problem I have tried to eliminate ...
0
votes
1answer
90 views

Permutations Question: Letter Arrangements with Restrictions

How many arrangements can be made of the letters in the word PHOTOGRAPH? What I did was, $8P5$ to find the number of arrangements between the two H's, then multiplied by 4! because the 5 letters and ...
3
votes
2answers
95 views

How to prove this assertion in $S_n$ for $n \geq 3$?

Let $n \geq 3$. Then there exists an element $f \in S_n$ such that $f \neq g^3$ for any element $g \in S_n$, where $S_n$ denotes the symmetric group on $n$ letters. How to establish whether this ...
0
votes
1answer
28 views

What is the relation between this subgroup and its conjugates?

Let $S$ be an infinite set, and let $A(S)$ denote the group of all bijections of $S$ onto itself. Let $M \subset A(S)$ be the set of all elements $f \in A(S)$ such that $f(s) \neq s$ for at most a ...
2
votes
3answers
154 views

What's the smallest exponent to give the identity in $S_n$?

Let $S_n$ denote the symmetric group on $n$ letters. We know that $\tau^{n!} = e$ for any element $\tau \in S_n,$ where $e$ denotes the identity element. Can we find a smaller positive integer $m$ ...
1
vote
2answers
73 views

Is there always a bijection mapping one element of an infinite set onto another?

Let $S$ be an infinite set, and let $s_1$, $s_2$ be any two distinct elements of $S$. Then how to determine whether or not there is always a bijection of $S$ onto itself that maps $s_1$ onto $s_2$? ...
1
vote
2answers
28 views

Combinations Question?

In how many ways can 6 different books be distributed between 2 students, provided that both students receive at least one book? Thanks for helping