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

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-1
votes
4answers
960 views

How many 4 digit numbers can be made with exactly 2 different digits

I need to find how many four digit numbers can be made using exactly 2 different digits. ALL DIGITS FROM 1 TO 9 CAN BE USED. NOT JUST 1 AND 0
0
votes
1answer
1k views

how many 2x2 matrices are invertible in mod p

I am trying to solve this problem for homework but unable to get anything. The question is to find the number of invertible 2x2 matrices in mod p? Each entery can bee from the set ...
0
votes
1answer
43 views

About definition of a sequences

The notion of sequence is basic notion in combinatorics and can be defined 1.Mapping from a finite set for example $$I_m=\{0,1,2,...,m-1\}$$ to an other set $X$ is finite sequence with terms in $X$ ...
6
votes
4answers
2k views

Exponential Generating Functions For Derangements

I have been introduced to the concept of exponential generating functions a few days ago. However, my understanding of them are still quite limited, and I would like to see some examples. Earlier this ...
0
votes
1answer
121 views

Different combinations of a string, with some restrictions

For the string "aaaaaabbbbbb" (that's $12$ characters with $6$ a's and $6$ b's), let's say that I want to find the number of unique possible strings that have exactly 4 of the same characters as the ...
1
vote
1answer
131 views

How many ways the electron can de-excite to ground state

This combinatorics problem arose from a question in physics: when excited from the 1st to the nth energy level, how many ways are there of making the path back to the 1st, allowing any combination of ...
1
vote
3answers
6k views

Number of subsets of a set having r elements

We have studied the standard way of ascertaining the total number of subsets of a set by using the concept of combinations ( or binomial coeffecients ). I came across an alternate derivation for this ...
1
vote
2answers
231 views

From $3$ red, $4$ green and $5$ yellow balls, how many selections consisting of $6$ balls are possible, if each color must be represented twice?

From $3$ red, $4$ green and $5$ yellow balls, how many selections consisting of $6$ balls are possible, if each color must be represented twice?
1
vote
2answers
190 views

Sum of four numbers less than a particular value

I have four positive numbers $a_1,\dots,a_4$, each less than $45$. How many different ways are there for $a_1+a_2+a_3+a_4<90$? I require different permutations i.e $a_1a_2a_3a_4$ is different from ...
32
votes
1answer
1k views

Is War necessarily finite?

War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules. A ...
0
votes
1answer
248 views

Six Unique numbers which generate unique sum during addition

I have a Six numbers which are like points that satisfy certain condition. If the condition is satisfied that point will be given or else 0 will be assigned as points. I am Storing the points in ...
2
votes
2answers
2k views

Finding the order of permutations in $S_8$

Consider the group $S_8$. What is the order of $\sigma = (4,5)(2,3,7)$ and $\tau = (1,4)(3,5,7,8)$? My book says I should just use a trick by the order of a permutation expressed as a product of ...
1
vote
0answers
162 views

secretary hiring algorithm

So I have to figure out that in the case of the secretary problem(http://en.wikipedia.org/wiki/Secretary_problem) what is the probability that I hire exactly one applicant over the course of going ...
2
votes
2answers
2k views

number of squares in a rectangle.

Given a rectangle of length a and width b(as shown in the figure).How many different squares of edge grater than 1 can be formed from using the cells inside . For example if a=2,b=2 ,then the ...
2
votes
2answers
178 views

Permutation and cycles

I need to determine the missing number to fulfill the following reproduction: $$\pi=\pmatrix{1&2&3&4&5&6&7&8&9\\3&5&9&4&1&2&6&7&8}$$ ...
2
votes
1answer
280 views

Number of possible matrices according to given rule?

You want to create a matrix of size MxN using only 0 & 1. But there is rule- ...
2
votes
3answers
66 views

Counting permutations of students standing in line

Say I have k students, four of them are Jack, Liz, Jenna and Tracy. I want to count the number of permutations in which Liz is standing left to Jack and Jenna is ...
2
votes
1answer
60 views

Number of combinations that do not contain all possible inputs

I have two balls: {A, B} and 3 slots. Each slot can contain one of the balls Balls can repeat, e.g. {A, A, B} is ok Order matters, e.g. {A, A, B} is not the same as {B, A, A} I want to know the ...
6
votes
3answers
656 views

Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily ...
0
votes
1answer
125 views

Permutations & Functions

This is an assignment question I received a week ago. A function $f:\{1, 2, \dots ,n\} \to \{1, 2, \dots, n\}$ which is a bijection is also called a permutation. Let $P_n$ be the set of all ...
2
votes
1answer
559 views

Rearranging Equations with Factorials?

Hullo! This is my first time using this site. I have just begun tutoring a Math 12 student and I'm a little rusty on factorials. The stuff I'm stuck on is why : n!/(n-2)! = n(n-1). I seem to be ...
1
vote
1answer
804 views

Find the nth permutation in dictionary order

What would be the most efficient way of solving the following problem? - A toy set contains blocks showing the numbers from 1 to 9. There are plenty of blocks showing each number and blocks showing ...
2
votes
0answers
47 views

Terminology: is there a term for one order being on a geodesic between two others in the Cayley graph?

Think about the graph whose nodes are total orders on a finite set, and whose edges connect orders that only differ on two elements. This is actually a Cayley graph of $S_n$, but I don't want to fix ...
1
vote
1answer
659 views

How many strings of length 12 can we compose using letters A, B, C, and D if every letter should appear at least once?

How many strings of length 12 can we compose using letters A,B,C, and D if every letter should appear at least once? can someone walk me through this? I believe using the concept of the sieve formula ...
3
votes
2answers
330 views

What is the number of rearrangements of the string AAABBBCCC that do not contain three consecutive letters of the same type?

Just a little combinatorial theory problem I am having trouble wrapping my head around. It has to do with rearrangement of a string of letters: Determine the number of rearrangements of the string ...
4
votes
2answers
306 views

Number of integer solutions to $x_1 + x_2 + x_3 = 0$ where each $x_i \geq- 5$?

I need to know how to find the number of possible integer solutions to the following problem. $$x_1 + x_2 + x_3 = 0 \text{ where }x_i \ge -5$$ Normally, I would do this problem by making it a ...
0
votes
1answer
36 views

Counting coin tosses

Here's a question I got as a homework assignment: A coin is tossed $k$ times. At some point the result was $T$, and then later on it was $H$. What are the number of possible permutations? So, ...
1
vote
1answer
231 views

permutations and the binomial coefficient

I have seen several times the use of "n choose k" in the left side of the permutations formula. However, this expression is usually referred to be used with combinations. Not that this change when or ...
0
votes
2answers
1k views

In how many inequivalent ways can 8 people be seated at a round table?

How many different arrangements are there of eight people seated at a round table, where two arrangements are considered the same if one can be obtained from the other by a rotation?
3
votes
2answers
126 views

The asymptotic behaviour for the probability that a random permutation in $S_n$ has order $2$.

Let $T_n$ be the number of elements of $S_n$ with order $1$ or $2$. It is well known that: $$ T_n = \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k}(2k-1)!! = ...
5
votes
1answer
130 views

Combinatorial properties of permutation groups

Let $P_n$ denote the set of pairs $(x,y)$ of permutations on $S_{2n}$, where each permutation is a product of $n$ disjoint cycles of length two. Let i and j be two fixed elements of the set $\{1,2, ...
3
votes
1answer
219 views

Ways $S_3$ can act on a set of 4 elements.

Describe all ways in which $S_3$ can operate on a set of four elements. My approach: This question can be broken down into: How many homomorphisms exist from $S_3$ to $S_4$. Say $\varphi : S_3 \to ...
2
votes
2answers
67 views

Permutation without a subword repeat

I am asked the following: Let m, n, and r be non-negative integers. How many distinct "words" are there consisting of m occurrences of the letter A, n occurrences of the letter B, r ...
1
vote
1answer
73 views

Question about cycle decompositions

The Question: Prove that (1 2) cannot be written as the product of 3 disjoint cycles. The Attempt: Suppose (1 2) has a cycle decomposition into 3 disjoint cycles $m_1, m_2$, and $m_3$. Then (1 2) = ...
0
votes
0answers
37 views

Permutational strategy for proving that a property holds for random sequences with given marginal $F(x)$

I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
2
votes
1answer
62 views

Permutation of Dresses [duplicate]

Possible Duplicate: In how many ways we can put $r$ distinct objects into $n$ baskets? How many number of ways I can wear four dresses for n days without wearing the same dress for two ...
0
votes
3answers
335 views

Probability of winning a pick 3 lotto game?

If you are given 3 standard 6-sided dice, and are asked to pick the order of the numbers that will appear; what is the probability that you will win, given that order DOES matter?
7
votes
2answers
572 views

Number of permutations which fixes a certain number of point

Given the set $N:=\{1,\cdots,n\}$, let $\pi$ be a permutation on $N$. We say $i \in \{1,\cdots,n\}$ is fixed by $g$ iff $\pi(i)=i.$ Denote the set of all permuations on $N$ by $S_n$. Define $f :~N ...
0
votes
1answer
136 views

Number of ways to write 5 digit number with restriction?

How many ways to write a 5 digit number so that every digit is scritctly greater than the digit on it's right ? How could we derive a formula for such a N digit number where N <= 9 ?
1
vote
1answer
159 views

Help needed in proving a theorem on a permutation that is the product of dis. cycles of prime length.

So we were given the following proof to do: Let $ p $ and $ q $ be distinct primes. Suppose $ \alpha $ is a permutation of $ S_n $ and suppose $ \alpha = \gamma_1 \gamma_2 $ where $ \gamma_1 $ and $ ...
2
votes
1answer
556 views

Permutation identities similar to $(7901234568 / 9876543210) \cdot 1234567890 = 0987654312$

It is well known that $9876543210/1234567890 = 109739369/13717421 = 8.0000000729...$ (See for example) Recently I posted at http://list.seqfan.eu/pipermail/seqfan/2012-October/010235.html my ...
1
vote
4answers
2k views

How many ways balls can be selected?

I am trying to solve In how many ways can 25 balls be selected from a bag containing 15 identical red balls, 20 identical blue balls and 25 identical green balls? Should not be the answer be - $$ ...
1
vote
2answers
189 views

Permutations of a sequence

"Given a set $\{1,\ 2,\ 3,\ 4\}$, how many sequences with a length of $4$ with entries from this set have exactly one entry equal to $1$?" Here is my work so far: $$X = \left\{\text{sequences with ...
0
votes
0answers
314 views

Find a Lipschitz constant

Please help me to find a Lipschitz constant. Let $S_n$ be a group of permutations of the set $\{1, \ldots, n\}$. Let $a=(a_1, \ldots, a_{2M})$ be a real valued vector with $n$ non-zeroes entries, $M ...
1
vote
3answers
74 views

Prove that $n!e-2$ $<$ $\displaystyle \sum_{k=1}^{n}(^{n}\textrm{P}_{k})$ $\leq$ $n!e-1$

Prove that $n!e-2$ $<$ $\sum_{k=1}^{n}(^{n}\textrm{P}_{k})$ $\leq$ $n!e-1$ where $^{n}\textrm{P}_k = n(n-1)\cdots(n-k+1)$ is the number of permutations of $k$ distinct objects from $n$ distinct ...
1
vote
2answers
135 views

Number of Sets problem

A class is attended by $n$ sophomores, $n$ juniors, and $n$ seniors. In how many ways can these students form $n$ sets of three people each if each set is to contain a sophomore, a junior, and a ...
8
votes
1answer
1k views

Centralizer of a given element in $S_n$?

It is known that any two disjoint cycles in $S_n$ commutes. Therefore, any $\pi\in S_n$ which is disjoint with $\sigma$ is in the centralizer of $\sigma$: $C_{S_n}(\sigma)$. Also $$ \sigma^i\pi\in ...
2
votes
2answers
238 views

Directed Graphs on Relations - Set Theory

These questions were from an assignment I had some time ago but the solutions were not provided. A permutation $P$ on a finite set $A$ is a binary relation with the property that for each $a∈A$, ...
1
vote
2answers
181 views

Shaking hands problem

$8$ friends meet at a restaurant. Each friend shakes hands with exactly $2$ other people. Find $N$, the number of possible lists of pairs of friends that shook hands with each other.
1
vote
1answer
700 views

Number of n digits having no same consecutive digits and same first and last digit

We have to form a number of n digits having digits from 1 to 9. Constraint is that first and last digit must be same and no two consecutive digits must be same. How many such number of n digits can be ...