What is Permutation: Definition and 275 Discussions
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.
Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.
The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.
Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3,1,2) mentioned above is described by the function
α
{\displaystyle \alpha }
defined as:
α
(
1
)
=
3
,
α
(
2
)
=
1
,
α
(
3
)
=
2
{\displaystyle \alpha (1)=3,\quad \alpha (2)=1,\quad \alpha (3)=2}
.The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition (performing two given rearrangements in succession), which results in another rearrangement. As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set
{
1
,
2
,
…
,
n
}
{\displaystyle \{1,2,\ldots ,n\}}
that are considered for studying permutations.
In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.
Let $N=\{ 1,2,\ldots ,n \}$.
As usual, take $S_n$ as the set of all bijections from $N$ to $N$.
Corresponding to each element $i$ in $N$ we have a set $E_i \subseteq N$ of "enemies" of $i$.
Denote $|E_i|=r_i$.
Its also given that:
1) $r_i \geq 0, \, \forall i \in N$
2) $r_1 + \ldots + r_n \leq...
Homework Statement
A crew of an 8 oar boat has to be selected from 11 people, out of which 5 can oar only on one side, 4 can oar only on the other side and 2 can oar on either sides, find in how many ways can the selection be done?
The Attempt at a Solution
What I did here is assumed...
Homework Statement
how many 4 digit numbers are there which do not contain more than 2 different numbers?
Homework Equations
The Attempt at a Solution
all can contain the same digit
any 3 out of the 4 places can be occupied by the same digit
and
any 2 out of the 4 places can...
Homework Statement
There are 15 players, out of which 4 players own one 4 seater car each, and the car will be driven by the owner himself. In how many ways can they be seated in the cars?
Homework Equations
(3n)!/(n!)3
The Attempt at a Solution
i know that i have to substitute...
Homework Statement
Evaluate the following expression:
\sum_{j}\sum_{k}\epsilon_{ijk}\delta_{jk}
Homework Equations
\delta_{ij} = [i = j]
The Attempt at a Solution
I don't have a solution attempt to this one yet, because somehow I completely missed out on what the permutation thing...
good day! i need to prove that the alternating group An is a normal subgroup of symmetric group, Sn, and i just want to know if my proving is correct.
we know that normal subgroup is subgroup where the right and left cosets coincides. but i got this equivalent definition of normal group from...
Hi. Consider two isomorphic state spaces \mathcal{E}(1) and \mathcal{E}(2). The first belongs to a proton, the second to an electron and they both have the same spin.
#Let B(1) be an observable defined on the first space, spanned by |1,u_{i}\rangle, eigenvectors of B(1) with eigenvalues b_i...
Homework Statement
How many way can a team lose 3 of their next 5 games?
How many ways can a team lose 2 of their next 5 games?
Why are the two answers the same?
Homework Equations
Permutation = nPr = n! / (n-r)!
Combination = nCr = nPr / r!
where,
n, r are non...
hey,
i have a quick question about G-sets and how the action of an element of the group on the set can be represented as a permutation of the elements of the set.
If you had a set S = {1,2,..,50} and a permutation σ: S→S where |σ|=30,
Does this mean that the minimum number of fixed...
All the proofs I have seen for this theorem uses the same argument: First prove that the identity permutation has even parity. Then let a be (one of) the first elements to appear in a transposition representation of a permutation in Sn. Then identify all the other transpositions in the...
Homework Statement
I have 6 cards with the digits 1,2,2,3,4 and 5 written on each of them. In how many ways can I arrange 3 out of the 6 cards in a row?
Homework Equations
The Attempt at a Solution
Case 1: Only one card with digit '2' two is taken, so 5P3 =60
Case 2: Both cards with digit...
Homework Statement http://www.xtremepapers.com/CIE/International%20A%20And%20AS%20Level/9709%20-%20Mathematics/9709_w03_qp_6.pdf
question 6 b ii
Homework Equations
The Attempt at a Solution
The concept is bothering me. the answer is 5! × 3! × 4C1. Why is it a permutation problem? Plus why do...
Homework Statement
Let Pn denotes the no. of ways of selecting 3 people out of 'n' sitting in a row if no two of them are
consecutive and Qn is the corresponding figure when they are in circle .
If Pn - Qn = 6 , then 'n' is equal to:
The Attempt at a Solution
Pn = (n-2) C 3
Qn =...
I already have the answer for this, but wondering if there's some efficient way to calculate this. There are 7 identical items, each with a equally random chance of being placed in 1 of 32 containers. What are the possible permutations?
00000000032 = 7 identical items into 1 of 32 distinct...
Homework Statement
I know what's happening with the problem, but I just hate myself drawing all possible arrangements, I think I am too stupid, I hope better solution.
Problems:
6 different Cookery books and 8 different History books are put on bookshelf. How many possible arrangements...
1. I think my professor gave me credit for this problem when he shouldn't have! I just want to understand for the test, so I want someone to verify for me.
2. How do you find out whether a given permutation is even or odd without factoring it into transpositions?
3. My answer was:
Factor it...
Homework Statement
4 books by Shakespeare, 2 books are Dickens and 3 by Conrad are chosen for the problem. The question is to find the number of ways in which the books can be arranged s.t. the 3 Conrad books are separated.
Homework Equations
The Attempt at a Solution
n(C separated)...
Hey,
I just have a small question regarding the conjugation of permutation groups.
Two permutations are conjugates iff they have the same cycle structure.
However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate...
Hey,
I just have a small question regarding the conjugation of permutation groups.
Two permutations are conjugates iff they have the same cycle structure.
However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate...
Homework Statement
Hello, it is a permutation / combination approach question, however, having thought about an hour i can't get any idea how it should be solved.
Question is
A coin is flipped n times, where n>=3
Find the number of ways to obtain
(i) exactly two heads
(ii) at...
Hi to everyone...
I would like to know which is the permutation 4661 of the elements G = {1,2,3,4,5,6,7}
I have search for the formula around the internet but I could not find it...
So I hope anyone can help about this solution.
Thank you...
and the result on the picture is like this as...
Homework Statement
I forget the exact expression of the questions. But the related details are exhaustive here.
it's about permutation and combination. By the way, I am not student, i am looking for explanation and understanding, not answers..
1. There are 6 people, namely A, B, C, D, E...
Use nested for loops to produce the following pattern of cyclic permutations of the English
alphabet:
abcde...yz
bcdef...za
cdef...zab
...
zabcde...xy
HINT: you may find the modulo (remainder) operator % useful.
I have an idea of how to do this but it would not use the modulo operator...
(Apologies for the lack of LaTeX formatting - I usually do my typesetting with MathType, but I only have Microsoft Equation Editor on this computer which doesn't have LaTeX export).
This isn't a homework question, but is problem I've discovered I'm facing after diving into another problem, but...
Four married couples attend a wedding dinner. One of the couples brought along two children. Find the number of ways in which these ten people can be seated round a table if each couple must sit together.
I need to know the logic and thinking process behind how the answer is derived.
What I...
can somebody help me to know the proper way to solve this kind of problem.
Five employees of a firm are ranked from 1 to 5
based on their ability to program a computer.
Three of these employees are selected to fill
equivalent programming jobs. If all possible
choices...
Homework Statement
How do I determine the parity of a permutation? I think my reasoning may be faulty.
By a theorem, an n-cycle is the product of (n-1) transpositions. For example, a 5 cycle can be written as 4 transpositions.
Now say I have a permutation written in cycle notation: (1...
there are n students in a class. They are selected in sets of m. A student says he is selected a times. find value of a in terms of m and n.
Total number of selections made are nCm.
After that nothing is striking in my mind.can you please help.
Homework Statement
A test consists of 5 pure math questions A, B, C, D, E and 6 statistics question F, G, H, I, J, K.
The examiners want to arrange all eleven questions in a random order such that a pure math question must be separated from another with exactly one statistics question...
Homework Statement
Suppose G is a group, H < G (H is a subgroup of G), and a is in G.
Prove that a is in H iff <a> is a subset of H.
Homework Equations
<a> is the set generated by a (a,aa,aa^-1,etc)
The Attempt at a Solution
For some reason this seems too easy:
1. Suppose a...
Homework Statement
The integers 1 through 6 appear on the six faces of a cube, one on each face. If three such cubes are rolled, what is the probability that the sum of the numbers on the top faces is 17 or 18? Homework Equations
Probability = # of desired outcomes / total # of outcomes
The...
I feel like I'm doing this correctly but my professor gave us an answer that is different from what I'm getting.
Composition of (12345)(12345)
Reading right to left I get (13524) but the answer he gives is (135)(24)
The odd thing is I'm getting the rest of these correct so I don't feel...
Good evening to all
I wonder if anyone can help please I have 8 switches b1,b2,b3,b4,b5,b6,b7,b8 I am trying to work out how many permutations I would have if I pressed any 4 of the 8 switches ie if I pressed b1,b3,b,7,b8 to make a circuit or it could be b3,b5,b6,b7 doing it by long hand I...
Hi,
I have this problem in which a word (lets say "ABERRATIONAL") is given. Now how many permutations (no unique words) can be made given that the 3 A's have to be next to each other?
My solution: regard the 3 A's as one block, so we have 12-3+1 = 10 letters. This gives 10! permutations...
Hi
here's a problem I am having.
Consider the hilbert space of two-variable complex functions \psi (x,y).
A permutation operator is defined by its action on \psi (x,y) as follows.
\hat{\pi} \psi (x,y) = \psi (y,x)
a) Verify that operator is linear and hermitian.
b) Show that...
I'm studying the S_n groups and I've been calculating a bunch of compositions of m-cycles. I haven't, however, been able to find a general pattern in how to do this for arbitrarily complicated cycles. Are there any general composition rules or is there no other way to find the compositions...
Homework Statement
Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G.
I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.
Homework Statement
What is the permutation matrix associated to the permutation of n indices defined by p(i) = n - i + 1? What is the cycle decomposition of p? What is it's sign?
Homework Equations
Prop. A permutation matrix P has a single 1 in each row and in each column, the rest of its...
Hey all,
I recently encountered a problem that I cannot seem to solve...
Let's say I have the following coins A, B, C, D and E, each one with an associated integer value such that:
A = 1
B = 2
C = 3
D = 4
E = 5
Let's say I can pick N of them, and order does matter (so ABC is not...
Let n ∈ N. Explain what is meant by saying that π is a permutation of n = {1, 2, . . . , n}.
The permutation A is given below in two line notation. Write A in disjoint cycle notation.
1 2 3 4 5
3 5 1 2 4
Find A^-1, writing your answer in disjoint cycle notation. Give two examples of...
Homework Statement
Let r be a positive integer. For any number x, let
(x)r = x(x-1)(x-2)...(x-r+1)
Show that
(-1/2)r = (-1)rr!2-2r(2r take r)
Homework Equations
by "2r take r" I mean what is usually denoted by (n / r) (written like a fraction but without the bar) and is calculated...
Homework Statement
In a test there were n questions. In the test 2n – i students gave wrong answers to at least i questions where i = 1, 2, 3, …, n. If the total number of wrong answers given is 2047, then n is
The Attempt at a Solution
I don't understand how to transform the second...
Hi all, I'm trying to create a program which allows the user to input two permutations of sizes <10 and then multiply two of these together.
I need some help on how to get two permutations to multiply each other in Fortran. A snippet of what I've written so far..
Module Permutations
Implicit...
My text defines an elementary permutation as follows:
Give 1 <= i < k, let e_i be the element of S_k (where S_k is the set of all permutations of {1,...,k}) defined by setting e_i (j) = j for j not equal to i,i+1; and e_i (i) = i + 1 and e_i (i + 1) = i.
It goes on to say that if f is in...
Homework Statement
So here's the problem. There is an elevator w/ 5 people equally likely to get off at any of the 7 floor.
What is the probability that no two passengers will get off the same floor.
Homework Equations
The probability of event should be P(E)= number of event...
1. I've only just scrapped through math in high school but recently I have started to take up an interest in the subject. I was listening to a math podcast and they posted a question to be answered the following episode. However, I cannot seem to track that one down and I really need an answer...