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

learn more… | top users | synonyms (1)

0
votes
0answers
26 views

Problem with random permutation and conditional probability

Let $\pi_1,...,\pi_n$ be a random permutation of numbers $1,...,n$. If you are told that $\pi_k > \pi_1,...,\pi_k > \pi_{k-1}$, what is the probability that $\pi_k = n$? What I've tried: Let ...
2
votes
2answers
28 views

Permutations with Condition

I have looked at this old problem in my textbook: How many permutations $\pi \in S_n (n \geq 3)$ meet the requirement: $\pi (1) < \pi (2) $ or $\pi (1) < \pi (3)$? I am not sure how to ...
0
votes
0answers
63 views

Examples of injective maps such as MTF (Book Stack). Set of such mappings.

Let $S \in \mathbb N$. Let $\mathfrak S_S$ be the set of permutations of size $S$. Consider map $f : \mathfrak S_S \times \{1,2,\ldots,S\} \to \mathfrak S_S$, such as $f(\alpha, \cdot) : ...
0
votes
0answers
37 views

For which integers $r$ is $\sigma ^r$ also a $k$-cycle? [duplicate]

Let $\sigma$ be a $k$-cycle in $S_n$. For which integers $r$, is $\sigma ^r$ also a $k$-cycle? I think I managed to prove that this is true iff $(k,r)=1$, but my proof was too long and not elegant ...
2
votes
1answer
92 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
0
votes
1answer
18 views

Hashing: Quadratic Probing

I have the following to prove, unfortunately I am not able to do so. Let h, h' be hash functions: $h(k,i) = (h'(k) + c_{1}i + c_{2}i^2)$ mod $m$. Show the following: if m is prime and $c_{2} \neq 0$ ...
1
vote
1answer
28 views

What is the distribution of cycle lengths in derangements? In particular, expected longest cycle.

There is a lot of information about expected cycle lengths in random permutations, but I'm having trouble adapting the arguments and calculations to the specific case of derangements - permutations in ...
1
vote
2answers
30 views

To prove an identity in permutation and combination.

I am trying to prove the following identity: ${n \choose 0}$ + ${n \choose 1}$ + $\ldots$ + $\frac{1}{2}{n \choose n/2}$ = $2^{n-1}$ where $n$ is even I know that I have to use few relations like ...
2
votes
1answer
36 views

Disjoint Cycles in a Cyclic Subgroup of $S_n$

If a permutation $\sigma$ $\in$ $S_n$, the permutation group of n elements, and $\sigma$ can be expressed as a product of disjoint cycles, is it necessary that the disjoint cycles be elements in ...
1
vote
3answers
65 views

In how many ways can a natural number be written as a sum of $2$ natural numbers?

For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.
-3
votes
0answers
35 views

12- In a standard deck of 52 cards, how many ways can you deal out 4 cards that are all black or all not face cards? [duplicate]

I did the sad mistake of taking math in summer school to boost my average. I am stuck on a few questions. In a standard deck of $52$ cards, how many ways can you deal out $4$ cards that are all ...
0
votes
1answer
54 views

Random permutation and isolated points on the line

Let $[n]=\{1,\dots,n\}$ be the (ordered) set of the $n$ first integers, and $\mathcal{S}_n$ denote the set of permutations of $[n]$. Let $1\leq k \leq \frac{n}{4}$ be an integer. If I draw uniformly ...
-1
votes
0answers
20 views

Bridge pairings

16 couples (call them "A" through "P") want to play bridge against each other once a week for six weeks. all 16 couples play every week and play every other couple at least once. 4 couples play in ...
0
votes
1answer
18 views

Success rate of a player trying to guess a bitstring with given constriants

For work at my university I try to solve a problem. I have a bit string with given length $len$ and count of active bit $active$ An example could be: 1001 0110 ...
-1
votes
2answers
35 views

Prove a cycle of length l is odd if l is even? [closed]

This is my first course on Group Theory. How do I go about proving this?
0
votes
1answer
46 views

Number of ways arranging entries of a tuple - combinations or permutations

Let $x=(x_1,x_2,\ldots,x_n)$ be an $n$-tuple where $n$ is even In how many ways we can arrange such that exactly half of the entries are even ? My attempt is : As we are talking about possible ...
2
votes
3answers
50 views

find all odd permutations $\sigma \in S_4$ such that $\sigma (123) \sigma^{-1} = (234) $

need help with this question... find all odd permutations $\sigma \in S_4$ such that $\sigma (123) \sigma^{-1} = (234) $ really have no idea how to approach this. thanks.
1
vote
1answer
33 views

Finding the number of combinations

A teacher distributes 7 books to 7 children (each student a books), on the next day she collects the books back and redistributes in such a way that each students get a new book. In how many ways can ...
2
votes
1answer
55 views

Find the probability of at least two vowels together when letters in word “AEINCB” are rearranged for all random permutations.

Find the probability of at least two vowels together when letters in word "AEINCB" are rearranged for all random permutations. What will be new probabilities when word is changed to either "AEINCBB" ...
1
vote
0answers
47 views

Can anyone explain these conclusions? Permutations, Symetric group…

The conclusions start off like this:I will highlight what is unclear in yellow. $sgn G$-sign of G permutation, $Ker$-kernel of a function Lets define the function: $\ \Phi$ like: $(\forall G \in ...
3
votes
2answers
45 views

Finding the number of ways of picking three cards

Problem: An urn has 10 red cards numbered 1 through 10 and 8 blue cards numbered 1 through 8. Three cards are randomly drawn, one at a time, without replacement. Find the number of ways to ...
3
votes
2answers
61 views

Finding how many bits of length n there are

So we are starting on the section of combinatorics in my discrete math class and our instructor gave us a simple problem to see if we understood what we learned that day. The problem consists of three ...
0
votes
2answers
45 views

question based on probability/permutation/combination

In a box containing $15$ apples, $6$ apples are rotten. Each day one apple is taken out from the box. What is the probability that after four days there are exactly $8$ apples in the box that are not ...
3
votes
1answer
36 views

Permutations of n objects taken r at a time ( r=1 to r=n ) where objects may be groups of same entities and it's sum

Given n objects where n1 objects are the same ,along with another group of n2 objects of same element etc.. such that Σni = n (i=1 to k). Assuming there are k groups of similar objects eg: in ...
-2
votes
1answer
45 views

Sum of all the numbers with the given numbers repeated

How to find the sum of all the numbers that can be formed using the digits 4,5,5,6,6,6 (This includes 4,5,6,45,46,54,55,....,666554). I knew that the answer is 39345806. I just need to know the method ...
2
votes
3answers
42 views

Permutations without repetitions (exclude repeated permutations) [duplicate]

The formula to calculate all permutations without repetitions of the set {1,2,3} is $\dfrac{n!}{(n-r)!}$ But how to calculate it if the set (or rather array in programming terms) includes repeated ...
0
votes
1answer
41 views

“quasi-increasing” permutation of a number

Call a permutation $a_1,a_2,\ldots,a_n$ quasi-increasing if $a_k\le a_{k+1}+2$ for each $1\le k\le n-1$. For example, $54321$ and $14253$ are quasi-increasing permutations of the integers ...
0
votes
1answer
17 views

Number of permutations with balanced middle element

Let $v$ be a permutation of $\{1,2,\cdots,2n+1\}$ where $n$ is odd, such that the middle element $v_{n+1}$ satisfies the following: the number of elements to the right of $v_{n+1}$ that are less than ...
0
votes
1answer
47 views

How many arrangements of 4 letters, with 3 of them being distinct, are there?

I read an example of the "Counting Principle" where we want to find the number of possible ways to rearrange 4 distinct letters chosen from the alphabet. The answer for this one makes sense. This is ...
0
votes
1answer
29 views

Arranging $2n$ objects in specific ways

There are $n$ objects $a_1, a_2, ... , a_n$ and another $n$ objects $b_1, b_2, ... , b_n$. We have to choose all the $2n$ objects such that $a_i$ is chosen before $a_{i+1}$ and $b_i$ is chosen before ...
0
votes
1answer
25 views

How to finding permutations where some elements repeat?

Sorry if my question is not mathematically correct. Please help me fix it if there is a better way to phrase it. So first of all, I know that if you have a list of numbers {1, 2, 3} then the number ...
4
votes
3answers
44 views

Permutation of Numbers

How many $5$ digit numbers can be formed from the integers $\{1,2,...,9\}$ if no digit can appear more than twice.(for example 41434 is not allowed) My approach is : Since, $max $ 2$ digits can ...
1
vote
2answers
28 views

Permutation of Indistinguishable Objects

How many number of two digit numbers can be formed using $\{4,5,6,6\}$ without repetition? I know that $\{45,46,54,56,65,64,66\}$ are the possible answers, but I am wondering if there is any formula ...
-1
votes
0answers
52 views

how to evaluate permutations of rubik's cube?

how to calculate total number of permutations of a rubik's cube , say, one face of the cube , specifically saying it is the blue face which is fixed , now what are the total number of permutations of ...
2
votes
1answer
50 views

Sum of permutations of a number

If I have a number (without any 0's) such as 1112334, how would I sum the permutations of its digits (excluding duplicates)? I am assuming there is a closed form involving factorials or combinatorics. ...
3
votes
1answer
38 views

Find the order of $\tau^{100}$

Let $\tau= \left( \begin{array}{ccc} ...
1
vote
2answers
42 views

Another kind of derangement?

I reading about derangements, and the following question came to my mind. Suppose in an office, there work 5 teams, each consisting of 1 head and 3 staff (so there is a total of 15 staff). If the ...
0
votes
1answer
26 views

Permutation of kronecker products

I would like to be able to compute a re-ordered kronecker product from the result of another kronecker product. For example, consider $$F=A\otimes B\otimes C\otimes D\otimes E$$ from the result F and ...
0
votes
0answers
43 views

The amount of unique structures that can be built from four different bases, without mirrored repetitions.

I'm trying to figure out the following problem. I have 4 different bases (A,C,G and T), and I'm trying to figure out how many possible unique structures I could build with them with different sequence ...
3
votes
1answer
37 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
4
votes
1answer
24 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
3
votes
1answer
43 views

How many permutations of the letters DANIEL do not begin with D or do not end with L?

How many permutations of the letters DANIEL do not begin with D or do not end with L? The correct answer is 696. This answer does not make sense as there are 120 (5!) ways the letters can be ...
0
votes
1answer
38 views

In how many ways can the committee be selected if the girls must include either Roberta or Priya but not both? [duplicate]

A committee of three boys and three girls is to be selected from a class of $14$ boys and $17$ girls. In how many ways can the committee be selected if the girls must include either Roberta or Priya ...
0
votes
2answers
38 views

Permutations and Combinations Tricky Question

In a photo there are three families (six Greens, four Browns, and seven Grays) arranged in a row. The Browns have had an argument so no Brown will stand next to another Brown. How many different ...
0
votes
2answers
46 views

Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
1
vote
2answers
77 views

How many ways are there of splitting twelve people into two groups of the same size?

Twelve people need to be split up into teams for a quiz. How many ways are there of splitting them into two groups of the same size? I did $12 C 6$, which gives $924$, however the answer is ...
2
votes
2answers
124 views

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet?

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet? I tried $5 C 1 + 5 C 2 + 5 C 3 + 5 C 4 = 30$, however, the answer is $20$. Any ...
3
votes
5answers
100 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
0
votes
2answers
22 views

Combination or Permutations - identification

Car $A$ can take $5$ passengers, car $B$ can take $6$ passengers, car C can take $2$ passengers. Find the number of ways that $11$ passengers and a couple to travel in the $3$ cars. $${13 \choose{5}} ...
1
vote
1answer
25 views

Inverse Permutations from $S_7$

Would someone mind giving an explanation of how to find the inverse permutation of: $(1 2 3 5 7)^{-1}$ in $S_7$? I am not quite understanding how to do this.