Permutations Definition and 287 Threads

Homework Statement Let t be an element of S be the cycle (1,2...k) of length k with k<=n. a) prove that if a is an element of S then ata^-1=(a(1),a(2),...,a(k)). Thus ata^-1 is a cycle of length k. b)let b be any cycle of length k. Prove there exists a permutation a an element of S such that...

Hi I'm having a little bit of trouble understanding about permutations, and how you multiply them? Say I have the permutations a = (45)(67), b = (46)(57) how do I multiply them? Thanks heaps! Jess

Hello, let's consider, for example, the Clifford algebra CL(2,0) and the following mapping f for an arbitrary multivector: a + b\mathbf{e_1}+c\mathbf{e_2}+d\mathbf{e_{12}} \longmapsto a\mathbf{e_{12}} + b\mathbf{e_1}+c\mathbf{e_2}+d For vector spaces R^n we can permute the coordinates of...

Homework Statement (1 2 3 4 5) (2 1 3 5 4) Write the bottom number as a product of transpositions Homework Equations The Attempt at a Solution 41352 41325 41235 42135 42315 24315 24351 21354 (2 4) (2 5) (2 3) (2 1)...

Homework Statement 1)Consider the permutation in S3 = ( 1 2 3 ) ( 1 2 3 ) NOTE: the two pairs of parenthesis are meant to be one pair that encases both rows Write as a product...

Homework Statement Show that every element in A(n)= set of even permutations, for n> or equal to 3 can be expressed as a 3-cycle or a product of three cycles. Homework Equations 3-cycle = (_ _ _). a permutation is a function from a set A to A that is bijective. The Attempt at a...

How many permutations (when objects are not all distinct) of size k can be created from a set of size N composed of n1, n2,n3,...,nr parts? When k = N this is easy and is equal to N!/(n1!n2!...nr!) The following question would be then how many combinations (when objects are not all distinct)...

Let A_n be a subgroup of S_n that includes all the even permutations. How many permutations of order 6 does A_6 include?

Homework Statement Solve for n. (14)_{n}P_{3}=_{n+2}P_{4} Homework Equations _{n}P_{r}=\frac{n!}{(n-r)!} The Attempt at a Solution First I write the problem with the equations written out \frac{14n!}{(n-3)!}=\frac{(n+2)!}{(n-2)!} I'm not quite sure how to isolate the n with all of the...

Homework Statement I am getting a little confused as to whether to use the number of permutations or the number of combinations as my sample space when it comes to probability problems, or whether it depends on the situation. Let's look at two of the examples from my text to help: Example 1...

Homework Statement Why are the permutations (1 4 2 6)(2 3 4 5) and (1 4 5 6)(2 3) equal? It seems to me as if the first pair of 4-cycles want to permute 4->2 and 4->5, yielding a contradiction, but I suspect I've misunderstood something about composite cycles. I suspect it has something...

Homework Statement If a multiple-choice test consists of 5 questions each with 4 possible answers of which only 1 is correct, (a) In how many different ways can a student check off one answer to each question? (b) In how many different ways can a student check off one answer to each...

1: Is a group of permutations basically the same as a group of functions? As far as I know, they have the same properties: associativity, identity function, and inverses. 2: I don't understand how you convert cyclic groups into product of disjoint cycles. A cyclic group (a b c d ... z) := a->b...

for a 3 x 3 matric of values a11 a12 a13 b21 b22 b23 c31 c32 c33 the determinant will be a11a22a33+a12a23a31+a13a21a32-a13a22a31-a12a21a33-a11a23a32 the last three are negative because they are odd permutations. The first three are even permutations A permutation apparently is...

Q in permutations and Combinations Out of a standerd 52 - card deck . how many 6-card will hearts and 2 clube ? my answer : 13 C 4 X 13 C 2 = 715 X 78 = 55.770 this is my answer please help me ...

Hi I want simple explanation of the Permutations and combinations and which one has condition and I want simple example to undersand it I want your help

How many permutations of the letters ABCDEFGH contain the string ABC? This is an example problem in my book, and the answer is 6! = 720. Could someone please explain to me the reasoning behind this (my book does a poor job explaining)? And would this reasoning apply if the string to be...

Homework Statement 1) What is the coefficient of x^43 in the expansion of [(2/x^2) − x3)^16? (2) What is the coefficient of x^14y^12 in the expansion of (3x − 2y)^26? Homework Equations Binomial Expansion The Attempt at a Solution For (1), I started out like this: (16...

Homework Statement prove the following natural numbers n and r. P(n-1,2) + 3P(n+1,2) = 2(2n^2 + 1) and P(n,r) = P(n-3,r-3) The Attempt at a Solution i honestly don't even know what this question is asking. this is a sort of handout of 3 questions our teacher gave us in which we...

The questions is pretty short... The source was a chat session and hence the simple sentence. I am stuck with this and thought that Homework Help doesn't attend out of the way questions and so I'm asking them here.

Homework Statement How many ways are there to rearrange the letters of the word “SUPERSTITIOUS” so that... (a) ...each T is immediately preceded by an S? For example, “STUERSTIPIOUS” has this property, while “SUPERSTITIOUS” does not. (b) ...the R appears between the...

Hi all, I've been having difficulty with the following question. Let P be a regular pentagon. Let R be the rotation of P by 72degrees anticlockwise and let F be the reflection of P in the vertical line of symmetry. Represent R and F by permutations and hence calculate: F R^2 F R F^3 R^3 F...

Homework Statement http: What is the largest number which is the order of an element of S_8? Write down an element of that order in disjoint cycle notation. Homework Equations The Attempt at a Solution To start with, I don't understand the wording of the question. When it refers...

this problem has been on my mind... how many permutations can you make with 30 'A's and 30 'B's or rather the same question, how many unique numbers can be made from 30 1s and 30 0s any ideas? (excluding permutations that look identical) thanks

Homework Statement Let \rho \inSym(n), p be prime, r be the remainder when n is divided by p (so 0\leqr<p and n=qp+r for some integer q). 1. Show that \rho^p = \iota iff the cycles of \rho all have lengths 1 or p. 2. Show that if \rho^p = \iota then |Supp(\rho)| is a multiple of p and...

Review problem. I have the answers. Don't know how to get them. Answers in brackets. Homework Statement Shortly after being put into service, some buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose a paricular city has 25 of these...

Homework Statement If \alpha,\beta\in S_n and if \alpha \beta = \beta \alpha, prove that \beta permutes those integers which are left fixed by \alpha. Show that \beta must be a power of \alpha when \alpha is a n-cycle. The other way round is easy to see, since if two cycles are disjoint...

Homework Statement can cycles a and b create cycle d? let cycle a = 123 cycle b=12345 cycle d = 12 i.e. can some combination of a and b = d Homework Equations only working in S5The Attempt at a Solution I have tried various different permutations, 23 over all, things like aba, ab(a^-1)...

[b]1. The problem statement, all variables and given/known Homework Statement 1)How many six letter subsets can you make if 2 are consonants and 4 are vowels. 2) There are 22 students in the student council and 4 people are to be elected. How many ways can they be elected if: a) there...

What is the number of bit permutations of the set {0,1}n and the number of circular right shifts of the set {0,1}n. I think the number of bit permuations is 2n, so is there 4 bit permutations here? Namely (0,0), (0,1), (1,0) and (1,1). And the right shift is just sending each element one space...

Homework Statement At a conference of 5 powers,each deligation consists of 3 members. If each delegation sits together, with the leader in the middle, in how many ways ca the members be arranged at a round table? Homework Equations No. of ways of arranging n objects around a...

Homework Statement Let x=(1,2)(3,4) \in S_{8}. Find an a \in S_{8} such that a-1xa=(5,6)(1,3) Homework Equations The Attempt at a Solution I have no idea how you go about finding the a. Help please.

Let us assume a dynamic system which has vector with 'n' components (which are non-negative integers from 1->n) at time t=1. In other words we have a permutation over 1->n at time t=1. Assuming time to be discrete, at any time time 't' , the system evolves such that there are 't' permutations...

Homework Statement A candidate sitting this paper is told to answer 5 of the 7 questions in section A, and 3 questions from the 5 options in section B, where not more than 2 questions from the same option can be chosen. Assuming that he answers 8 questions altogether, find how many different...

I've been thinking about this one for over a week now. Does anyone have any smart way of counting the permutations of {1,2,3,...,n} that are of the form a_1< a_2 > a_3 < ... > a_n? You notice that there are (n-1) ways to choose the first elt. since the first choice cannot be n. And the...

Changed to a different question: Can anyone provide a proof for the order of alternating groups |An| = ½(n!)?

Hi to all of you guys here… A friend of mine gave me: 1). A paper with a table of 350 rows x 284 columns, which each cell contains of a single number from 0 to 9. This table didn’t typed yet into .xls file. It will be like table on sheet 5 of file Enigma-2.xls if it has. Since here I can’t...

How many permutations can be made out of the letters of the word STATISTICS?

Homework Statement Compute the expression shown for the permutations 1.\left|<\phi>\right| 2..\left|<\tau^2>\right| 3.\phi^{100} where: \phi= top row:1, 2 , 3 ,4 , 5 ,6 bottom row: 3,1, 4,5,6,2 \tau = top row: 1,2,3,4,5,6 bottom...

Homework Statement 9 people, 2 cars can hold 5 people each. only 3 people have licenses. Homework Equations How many different ways can the people be seated into cars? The Attempt at a Solution There are nine people. So number of seats is not a problem. We must find out how...

Homework Statement 7 distinct flags are hoisted in a post. Find the number of ways of arranging them if 2 of the flags must be separated. (The answer is 3600) Homework Equations Permutation: n! Circular Permutation: (n-1)! The Attempt at a Solution 1: Flag 1, 2: Flag 2: The...

Given this permutations {1,2...,n}, prove that the directions of 1 and 2 never change. Proof: When generating permutations, one starts with everything having a left facing arrow. In order to determine what is mobile, the arrow must be pointing towards a smaller integer. 1 points to nothing...

Homework Statement a = (162)(45) b = (123)(46) c = (1362) Find: ab Homework Equations The Attempt at a Solution k, so for a i have this: | 1 2 4 5 6 | | 6 1 5 4 2 | and for b i have this: | 1 2 3 4 6 | | 2 3 1 6 4 | so when I started doing the multiplication I...

Homework Statement A family of nine has two vehicles, each of which can hold a maximum of five people. Homework Equations How many different ways are there of allocating people to the cars? If three of the memebers of the family only have a drivers licence, how many different ways...

Homework Statement Water is poured into the top bucket of a triangular stack of 2-L buckets. When each bucket is full, the water overflows equally on both sides into the buckets immediately below. How much water will have been poured into the top bucket when at least one of the buckets in the...

If you have a permutation writtein disjoint cycle notation ( I attatched it ) what's the minimum positive integer k such that t^k =Identity t is tau

I'm looking at a card shuffle. And in shuffle would be the permutation (1, 2, 3, ..., n, n+1, n+2, n+3, ...2n) to (2, 4, 6, ..., 2n 1, 3, 5, ...2n-1) I know that it would take 52 perfect shuffles to get the deck of cards back in the original order. I think that I'm supposed to show this using...

What's the difference between these two: 1) The number of permutations of n distinct objects taken r at a time is \frac{n!}{(n-r)!} and 2) The number of combinations of n distinct objects taken r at a time is \frac{n!}{r!(n-r)!} ?

Homework Statement Not really a problem, just my understanding. What is the difference between them? I know the formulae are different. They seem to be the same thing, that is, n objects taken r at a time. Any help in clarifying this would be appreciated.

This seems like it should be easy, but my combinatorics is (are?) a little rusty. I want to know how many permutations of a given length leave at least one element unchanged, and how many leave exactly one unchanged. Started wondering about it when picking names out of a hat for a secret...