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

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Number of seating arrangements in 5 cars

An exercise from Introductory Combinatorics by Richard A.Brualdi: A roller coaster has five cars, each containing four seats, two in front and two in back. There are 20 people ready for a ride. ...
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1answer
16 views

number of elements in unsortet case

I have a group M with Mn different elements. How many unique combinations can I make out of this in an n digit system when order is no importance. For example if M = {1 2} & n = 3 ...
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1answer
60 views

How many ways are there to place these books on the shelves?

You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.) I want to say $7^5$ since there are 7 possible shelves and ...
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1answer
28 views

triangles and lines

There are 12 points in a plane. If 4 of them are on a straight line and no other 3 points are on a straight line, then find the difference between the number of triangles and the number of straight ...
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1answer
23 views

arrangement of balls in bowls

There are five bowls numbered $1$ to $5$. There are $5$ green balls and $6$ black balls. Each bowl is to be filled by either a green or black ball and no two adjacent bowls can be filled by green ...
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3answers
50 views

Writing a permutation group in 2 row notation

I have a permutation group in $S_7$, namely: $$(12345)(137)(56)$$ How do I write this in two row notation? I am to write it as disjoint cycles and then as transpositions but I feel better working in ...
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1answer
35 views

how can you count number of digits used in numbers from -2^127 to (2^(127) - 1)

There are numbers from -2^127 to (2^127)-1. I want to count the number of digits used in all the numbers. For example If I have numbers from -100 to 100 then number of digits used is $2*(1*9 + ...
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2answers
41 views

Kernel of $\phi:G \rightarrow \operatorname{Sym}(S)$ Group actions

$\operatorname{Sym}(S) == \text{All permutations of the set }S$. Prove $\ker(\phi)=\bigcap_{x\in S}G_x$ where $G_x$ is the stabilizer of $x$. Let $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
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1answer
5k views

How many possible combinations in 8 character password?

I need to calculate the possible combinations for 8 characters password. The password must contain at least one of the following: (lower case letters, upper case letters, digits, punctuations, special ...
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1answer
54 views

Number of permutations of AABBBCC, taking 7 letters at a time, when repititions are allowed

What is the number of permutations of the word AABBBCC, taking 7 letters at a time, repetitions being allowed? I think it should be $3^7$, but I can't see why. Also what would be the number of ...
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0answers
29 views

How many commutative block ciphers are there?

Let $K$ and $M$ and be two finite sets. Let $(G,\circ)$ be the group of permutations over $M$ under composition. Let a (implicitly: block) cipher with key in $K$ and message in $M$ be any application ...
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2answers
35 views

Difference between permutations

Given the following: 1) Is it wrong to say (1 2 4) (5 3) = (1 2 4) (5 3) or = (3 5) (1 2 3) ? 2) What is meant by ( 1 2 3 4 5 ) and 1 2 3 4 5 ? And why are they not equal? Thanks!
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1answer
22 views

Permutation Multiplication (easy)

Given α◦β=(1532)(14)(35) How do we get from the given to = ( 1 4 5 2 ) ( 3 ) = ( 1 4 5 2 ) = (4 1 3 5 2) ? Thanks
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1answer
83 views

List all the permutations of {1,2,3,4}. Which are even, and which are odd?

The answer is: There are 24 permutations. The 12 even permutations are: id , (1 2 3 4) , (1 3 2 4) , (1 4 2 3) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3). The ...
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2answers
109 views

Powers of permutation matrices.

Let $P$ be a permutation matrix obtained by the identity matrix by switching 2 rows $n$ times, (with no two rows switched more than one time). How to show that $$P^{\ n+1} = I$$? Is it true that, ...
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1answer
41 views

number of ways of constructing $n\times2$ rectangle from a $1\times2$ rectangle

You are given $1\times2$ rectangles and you have to construct an $n\times2$ rectangle from it. Tell the number of ways of constructing $n\times2$ rectangle from a $1\times2$ rectangle
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2answers
113 views

A man, woman, boy, girl, cat, and dog are walking down a path..

I'm hoping someone can explain how this works. The problem: A man, woman, boy, girl, cat, and dog are walking down a path in single file. How many ways can this happen if the dog is between the man ...
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0answers
48 views

Composing permutations in factorial notation

Given two permutations $p_1$ and $p_2$ in factorial notation, is there a direct algorithm which computes their composition directly, i.e. without translating to a different notation or via computing ...
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1answer
172 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
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0answers
30 views

Number of permutations with double restriction

Task is as follows: Let's have 6 element set, there are obviously $6!$ permutations of this set, but there are two restrictions: element 1 and 2 have to be in one cycle and element 3 can't be with 1 ...
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2answers
34 views

How many good words are there?

A “good” word is any seven letter word consisting of letters from $\{A,B,C\}$ (some letters may be absent and some letter can be present more than once), with the restriction that $A$ cannot be ...
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1answer
56 views

Why is this the method to getting transpositions from disjoint cycles?

I have the disjoint cycle: $$(156)(2437).$$ Apparently the "method" would get us: $$(1,6)(1,5)(2,7)(2,3)(2,4).$$ Basically you take the first number, and put it as a transposition of the last number ...
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1answer
75 views

How many ways can six of the letters of the word ALGORITHM be selected and written in a row if the first letter must be A?

As the title states, the question is: "How many ways can six of the letters of the word ALGORITHM be selected and written in a row if the first letter must be A?" I don't really get what the problem ...
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3answers
45 views

Combination with repeats

I've been stuck for a while on this question and haven't found applicable resources. I have 10 choices and can select 3 at a time. I am allowed to repeat choices (combination), but the challenge is ...
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1answer
27 views

Permutation and combination regarding two people together

You have 11 friends and can invite 5 to dinner 1) in how many ways if two of the friends married and will not attend separately? 2) in how many ways if two of them are not on speaking terms and will ...
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2answers
23 views

Permutation of numbers and function

Let $N(x)$ denote the number of terminating zeroes of the number x. If A,B,C is a permutation of the numbers $211^{19}+9$, $9^{101}-9$, and $19^{111}-9$ such that $N(A)<N(B)<N(C)$, determine the ...
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1answer
17 views

Prove that cyclic index of this operation can be expressed by formula

Let $T_1$ and $T_2$ be disjoint finite sets and let $G_1$ and $G_2$ be, respectively, some groups of permutations of this sets. Direct sum $G_1 \bigotimes G_2$ acts on $T_1 \cup T_2$: $$ \langle ...
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1answer
86 views

How to assign values to letters to create unique values per word when all letters are added together?

I'm writing a program to match anagrams in order to practice coding. One way I want to try this is to assign values to letters such that adding up the letters in the individual words creates a unique ...
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2answers
58 views

Permutations of 7 numbers less than 50

How many solutions does the equation $a + b + c + d + e + f + g ≤ 50$ have if each variable must be a non-negative integer? I feel that the answer is $50 \choose 7$? but that seems far too simple.. ...
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1answer
21 views

Clarification on the variables used in combination formula

Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at (10, 15). The ant always chooses to walk exactly one unit either up or to the right (towards his destination) ...
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1answer
47 views

What is the value of $1 + {{}^nP_2}/2 +{{}^nP_3}/3 + ~… ~ {}^nP_n/n $

What is the value of $1 + {{}^nP_2}/2 +{{}^nP_3}/3 + ~........... ~ {}^nP_n/n $ ${}^nP_r = \frac {n!} {(n-r)!}$ Attempt: $1 + {{}^nP_2}/2 +{{}^nP_3}/3 + ~........... ~ {}^nP_n/n $ $= 1 + \frac ...
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3answers
112 views

Permutations and Combinations

A fair $6$-sided die is rolled 5 times and the result is recorded for each roll. How many different results are possible? Of the possible results, in how many ways can there be a result containing ...
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1answer
54 views

Finding the smallest positive integer $n$ such that $S_n$ contains an element of order 60.

I am trying to find the smallest positive integer $n$ such that $S_n$ contains an element of order 60. I know that every permutation in $S_n$ can be expressed as the product of disjoint cycles, and I ...
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1answer
53 views

Left Regular Representations of a Group

The definition in my book gives it as $\lambda_x$ such that $\lambda_xg=xg$.I understand this much. But my question asks to find the left regular representation of $\mathbb{Z}_3$ in $S_3$. I don't ...
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1answer
24 views

Permutations of a sequence of words

I've been given a question in class and I just wanted to confirm the answer ...
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1answer
92 views

Find the number of elements in the set: $A=\{\sigma\in S_4 |\thinspace \sigma\thinspace(3)=3\}$

Find the number of elements in the set: $A=\{\sigma\in S_4 |\thinspace \sigma\thinspace(3)=3\}$ I know that this would be $3!=6$. But are these the correct elements? $$ \{e, (12), (24), (14), (142), ...
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1answer
33 views

Combinations formula

What is the no. of ways to distribute N identical objects among two persons such that at every instant first person gets more than the second person? My approach is : For N=1 ans=1 For N=2 ans=1 For ...
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0answers
36 views

Using permutations to determine whether given arrangements of the “eight puzzle” are possible.

The "eight puzzle" is a reduced 3x3 version of the 4x4 "fifteen puzzle", which is a very simple game which involves sliding 15 numbered tiles around 16 places, with one free space always being ...
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1answer
47 views

Showing that there is a permutation $\rho$ that fixes a number that $\sigma$ moves when $\rho \sigma \rho^{-1}=\sigma^{-1}$

Doing an assignment, getting a bit frustrated with this exercise, would really appreciate some help. The first exercise explains what $\sigma$ and $\rho$ are: Let $\sigma$ be the $r$-cycle ...
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2answers
107 views

Is there a solution to this Seating Plan problem?

So a colleague asked me for some Help on an interesting Problem, which we both couldn't find the optimal answer for. The event which needed it is already in the past, so this is just me trying to ...
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1answer
99 views

How many ways to paint a board with 2 colors..

You got a fence, you need to paint the boards with black and white, but can not have 3 or more boards same color in a row. how many ways do you have?
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1answer
47 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
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Permutation and combinatorics problem

How many numbers between 10 and 1000 can be formed using digits 3,4,5,7? Should we first find number of 2 digit numbers and then find number of three digit numbers and add them
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22 views

every k cyclic is a product of at least k-1 distinct tranpositions

There is a theorem says if $A$ in $S_n$ is a $k$ cycle, and $A = a_1 a_2 a_3 \dots a_m$, where $a_i$ are transpositions, then $m \geq k-1$. But how to show there are at least $k-1$ distinct ...
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3answers
200 views

In how many ways can the couples sit?

$4$ married couples are to be seated on a circular table with $8$ identical seats. In how many ways can they be seated so that (i) males and females sit alternately and (ii) no husband sits adjacent ...
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3answers
411 views

How many different 4 digit combinations will include at least one 7, assuming numbers cannot repeat

I cannot get the correct answer - $2016$. What I have tried so far is thus: the number $7$ can occur $1, 2, 3,$ or $4$ times. Since it is a combination we do not care if the number starts with zero ...
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3answers
266 views

What does “order matters” regarding permutations refer to?

I psychoanalyze EVERYTHING and permutations/combinations are frustrating me. Sorry for posting so many questions lately but I really appreciate all of the help! Ok so I know the permutation formula: ...
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1answer
45 views

Can someone please explain the reasoning for this permutation problem?

I'm studying and reading through my discrete math book.. I seemed to be grasping the idea of permutations, but I don't understand how the solution for this particular problem came to be. Question: ...
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0answers
29 views

Arrangements of numbers when each can have multiple values?

I feel as though this is fairly straightforward, but I can't figure it out. If I have $n$ numbers, clearly these can be arranged in $n!$ ways. But if each of the $n$ numbers can have a value $v, ...
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2answers
143 views

Permutations of a letter sequence

I have been given a question about (i think) permutations, it asks to find how many x letter sequences can be made out of a word. I am getting confused because it first asks how many different ...