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

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Maximum winner matches

N players take part in tennis championship. In every match loser is out. Two players can play a game if in that moment the difference of played games of that two players is not more than 1. They are ...
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45 views

Is there away to swap adjacent items in a sequence, such that each permutation occurs once?

Let's us say you have a finite sequence of things. Some are identical, some distinct. For example: $$\langle 2,5,7,2,3\rangle$$ Now, under what conditions can each permutation of a sequence be ...
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How many permutations of all the letters in the word ARMADILLO begin with letter A?

I know that there are 9 total letters and there are three A`s and two L`s. Is the answer just $9!/(3!*2!)$? Thanks for any help.
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112 views

Do cyclic permutations of rows and column entries generate all permutations?

Background: I am interested in the group of permutations of the entries of a general $m\times n$ matrix. In particular, I am interested in (1) interesting sets of simple generators for this group ...
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161 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
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If $F$ is a field, show the following function is a permutation

Let $F$ be a field. Show that the function $a\rightarrow a^{-1}$ is a permutation of $F\{0_F\}$ So I know that if it is indeed a permutation, then it is one-to-one and onto. Also, For every $a$,$b$ ...
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A ~Class~ of Permutations Appears to Always Have Order <= N

Details: Sometime in high school, we had enough time before the next class to be bored - so I walked up to the board (not having the slightest idea what I would do when I got there), &: wrote a ...
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Possible number of permutations

I've generally been seeing the formula nPr = n!/(n-r)! to calculate the number of permutations. But this is the case only when elements cannot be repeated, am I correct? If elements can be repeated, ...
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How many different sequences in this event?

There are $10$ students in the class. Four of these students will be selected randomly to participate in the math contest. Then, how many different sequences can four students be selected? I thought ...
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22 views

Suppose we have a chair with $n$ legs and it stands with all legs irrespective of floor's quality i.e smoothness,evenness, then what is $n$?

Suppose we have a chair with $n$ legs and it stands with all legs irrespective of floor's quality i.e smoothness,evenness, then what is $n$? 1)2 2)3 3)4 4)5 I have no clue what they want to say, ...
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Generalize the number of total overall hops possible in a line network

Suppose I have a network of computers arranged in a line, like in the image I made below. I want to know the total number of hops possible. For example, $A$ gets to $B$ in $1$ hop, to $C$ in $2$ and ...
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How many configurations

If I have twelve objects, each of which can be in one of three states (say on, off, on/off), how many ways can the group of twelve be configured -- by changing the state of each object?
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1answer
25 views

Number of k-colorings as a fraction of all possible ways to color a graph

I have a graph with $n$ verices, and I want to compute the number of ways to color the graph (with no adjacent vertices having the same color) using anywhere between $1$ and $n$ colors. This number ...
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2answers
28 views

Find the number of different teams that can be selected if the team contains all $4$ women?

A team of $6$ people is to be selected from $8$ men and $4$ women. Find the number of different teams that can be selected if: (i) there are no restrictions, (I solved this using ${n\choose ...
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41 views

Find how many of these $4$-digit numbers are even. [closed]

(a) (i) Find how many different $4$-digit numbers can be formed from the digits $1, 3, 5, 6, 8$ and $9$ if each digit may be used only once. I did this; the answer is $360$; I used ...
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58 views

password attempts

The password must have a length of 8 and be composed of lower case letters and/or numbers (letters and numbers can be repeated). Suppose someone wants to access your online bank account. ...
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1answer
27 views

Permutation that gives a sequence of non-negative partial sums

Given a sequence $(a_i)_1^n$ of real numbers that sum to $0$. There are $n$ circular permutations. $\begin{array}{}\sigma_1 = &(a_1 &a_2 &\cdots &a_{n-1} &a_n)\\ \sigma_2 = ...
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25 views

number of possible combinations of events

I am not a mathematician but I'm giving a lecture on the scope of a number of people in a given group to interact. There are 40 people in an affinity group Each person has 10 unique items of ...
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27 views

How to calculate the HHG (Heller Heller Gorfine) correlation

HHG (A consistent multivariate test of association based on ranks of distances) is introduced in: Heller, R., Heller, Y., & Gorfine, M. (2012b). A consistent multivariate test of association ...
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3answers
36 views

Permutations with repetitions - which is k and which is n?

I am learning permutations with repetitions, and working with the formula that $P(n,k)=n^k$. I understand the logic that with repetitions, we multiply $n$ by itself $k$ times. But in different ...
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46 views

Permutation Formula Question

The problem that I am having trouble figuring out is: In how many ways can give five men and five girls be seated at a round table so that each girl is between two men? I know that the formula for ...
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29 views

probability of finding a small sequence within a larger sequence

I'm wondering how to define the probability of a long string LS (using 26 letters alphabet) to contain a smaller string ss. Right know I have something like this. Number of LS containing ss: 26^( ...
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3 letter combination

Please excuse me if this question is too juvenile for this forum. Actually weirdly I cant figure this out. I have 3 letters $A, B, C$ I need a formula that gives me all the possible arrangements for ...
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30 views

combinations and permutations - choosing when there's a limit

A kid can choose 7 out of 12 donuts to eat. How many ways can he do this if he must choose at exactly 3 of the first 5? Similarly, how many combinations are there if he must choose at least 3 of the ...
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Combinations and permutations when separating into groups

If I have 30 people and I want to form them into 3 groups. One of size 10, one of size 5 and one of size 15. How many ways can I do this? Similarly what if I have 15 boys and 15 girls and the 10 ...
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58 views

Write the permutation as product of cycles

Let the permutation $$\sigma=\bigl(\begin{smallmatrix} 1 & 2 &3 &4 &5 &6 & 7 &8 & 9 & 10\\ 3& 4 & 5 &6 & 7 & 2 & 1 & 10 & 9 ...
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combination GRE problem 25

An appliance's model number consists of three alphanumeric characters. The first character must be one of 24 permissible letters of the alphabet. The next character is numeric, a digit from 1 to 9. ...
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List vs Permutation representation.

If we have a list xx:=[3,7,17,16,15]; and I want to convert this list into permutation like xx:=(3,7,17,16,15); I used PermList(xx) for this but it doesn't convert it in the required form. How can ...
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Comparing test statistics and/or $p$-values from multiple permutation (Mantel) tests on non-independent data

I am comparing the relationship between genetic and geographic distance of individuals in a wild animal population. The hypothesis is that individuals with higher genetic relatedness establish home ...
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Find k such that, $\displaystyle\frac{n^{\underline k}}{n^k}<a$

How to find k here $\displaystyle\frac{n^{\underline k}}{n^k}<a$ ? With, $n^{\underline k}=n\cdot(n-1)\cdot...(n-k+1)$ and $(n>k,\ a>0)$ Of course if $k\approx n/2$ the inequality ...
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show permutation graph is a perfect graph

I need to prove that permutation graph is a perfect graph. I've shown that for any $i<j$, $\{i,j\}$ connected iff $f(i)>f(j)$. In addition, I know that the complement of a permutation graph is a ...
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Dividing a set of n elements into k disjoint subsets.

I have been able to do the 1st part. I have not been able to prove the 2nd part. My attempt to the solution :- I took $k$ groups $ a_1, a_2, a_3…, a_k $ Let $a_1$ group has $b_1$ similarly so ...
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19 views

Number of subset-permutations for a set

Suppose you have a set {a, b, c}. There are 3! = 6 permutations. Subsets {a, b}, ...
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51 views

What is the meaning of nCk X nPk?

I am trying to understand the bulls and cows document, Page 6 , equivalences . Can someone please tell me what author means when he says nCk x nPk like 4P0 X 4C0 , 4P1 X 4C1 ?
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|x|+|y|+|z|=15. How many integer solutions do exist?

I tried to truncate the problem to $|x|+|y|=15$, which is giving me the answer to be $60$ solutions. However, I am trying to apply beggar method and everything shatters. How should I proceed? Can I ...
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35 views

Probability of boys ahead girls

There are three boys and two girls in a Queue. What is the Probability that number of boys ahead of every girl is at least one more than number of girls ahead of her. My Try: Let $B1$,$B2$,$B3$ are ...
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Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
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37 views

A Circular permutation problem

i've this problem that's making me crazy. I've 12 ball, 4 of these are red, the others are white. What's the probability to obtain a circular sequences of balls in which red balls are not adjacent? ...
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Mean and Variance of Correct Answers for various ways of Multiple Choice Questions

I picked this question out of the blue while thinking about multiple choice questions. Consider I set, say $8$ multiple choice questions. Now, I want to find out, out of the three methods below, what ...
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1answer
36 views

Find the permutation

This is part of an exercise I did on an assignment but I am having trouble remembering how to complete the exercise (even though I got full marks on my assignment). Let $P_1=(3\,4\,1\,2\,5), ...
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69 views

Theorem in finite fields fails in my example

I need to understand the following theorem, so i did an example. But i realized that i don't get everything in the finite field theory. can somebody check the example and say where the mistake is? ...
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A common combinatorics problem

Suppose the final result of a football match is 5−4, the home team winning. If the home team scored first and kept the lead until the end, in how many different orders could the goals have been scored? I ...
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How to get all permutations of a variable-length word

I need to find all permutations of a set of letters (word) with following parameters: Word lengths $\ell = [1, 20]$ Alphabet $A = \{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t\} \Rightarrow \lvert ...
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18 views

Given a string containing 'a' and 'b', Find all permutations starting with 'a'

Given a string contains only a and b where a comes M times and b comes N times in the string. I need to compute all the permutation of the string starting with a. What is the most efficient way? Can ...
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Permutation and Combination Puzzle - Spy Keypad

Keypad 1 2 3 4 5 6 7 8 9 J. Bond has to break into the headquarters of an evil organization and steal important documents. The documents are in a safe that can ...
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Probability: What is wrong with my approaches?

$4$ out of $20$ balls are black. The others are white. If I pick $2$ balls out of those $20$ randomly, what is the probability that at least one of them is white? I could do this using a ...
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Dropping letters in post boxes

In how many different ways can 5 letters be dropped in 3 different post boxes if any number of letters can be dropped in all of the post boxes?
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Fill $8$ boxes with $60$ items

I have $8$ boxes and $60$ items: how many ways can I fill the boxes so that The order of the items in each box does not matter It does not matter which boxes are filled with which items. In other ...
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20 views

Ordered Partitions

I'm having trouble figuring out exactly how one would get the answer for a question such as: "how many ways can 9 concert tickets be divided between 4 concertgoers, such that one person receives 3 ...
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81 views

Probability: deep understanding of permutations and combinations.

Look at this problem: If one is to throw $2$ dice, what is the probability that he/she gets a total of exactly $4$ dots? Well, the amount of the events that satisfy the condition is $3$ as we ...