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

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Permutations product

Can anyone please help me compute $513642798 \times 971265384$ attempt: I start with the right permutation, So $9 \to 7 \to 9 = (97)$ however, after that do i go to $7 \to 1 \to 3$ ...? I am ...
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4answers
88 views

Permutation on word if E,F,G have to stay in order

Im stuck on a problem which I have answered and need help to verifiy if I have done/understood it correctly. Problem If we have the following string: ...
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1answer
44 views

permutations of “optimization”

How many permutations of the word optimization are there? I get confused with the repetition of the letters. If all 12 letters were distinct, then we would have 12! Because 4 letters are repeated, I ...
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1answer
298 views

Find total perfect combinations

Suppose we are given N numbers and a value K. Now we can interchange the positions of numbers to form different combinations, where if there are two same numbers then their arrangments will be ...
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2answers
46 views

combinatorics - permutations question, possibly with pigeon hole

Let $A \in Mat_n(\mathbb R)$ such that $\forall i,j: a_{ij}\geq 0$ We are given: $$\forall j: \sum_{i=1}^n a_{ij}=\sum_{i=1}^n a_{ji}=1$$ show there's a permutation $\pi \in S_n$ such that $$\forall ...
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1answer
174 views

Permutation of a 4 character string made up of letters and numbers

This is a straightforward question but I didn't pay attention in school. I want to know how many permutations there are for a 4 charcter string made up of numbers and letters. After a quick look ...
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0answers
44 views

Notation for number of tensor permutations

I have a tensor (or set if you will) that consists of N elements, and each element has a limited number of values it can take. For example: ...
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2answers
208 views

Number of bitstrings with $000$ as substring

I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$? Now there are many ways to go about this but if I choose starting ...
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1answer
75 views

Combinatorics - inclusion exclusion, check my answer

It's my second try to solve the question I posted here Combinatorics question - How many different ways to change sitting order I got some really good advice, but no one said the answer, I solved it, ...
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1answer
85 views

permutations of a 3 object

Help please, Exercise: Work out the full multiplication table for the set of permutations of three objects. I know there are 6 permutations. (1,2,3), (2,3,1), (3,1,2), (2,1,3), (1,3,2) and (3,2,1). ...
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126 views

Probability and combination/permutation help? [closed]

So, I need some help on a few problems. This isn't my homework. I'm preparing for a test and these are just some of the practice problems. I have the answer for them but I don't understand how to get ...
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1answer
118 views

proving bitstrings [duplicate]

1.Let it be $a_n$ the number of bitstrings which contain 000 How would I prove that for $n\ge4$: $$a_n = a_{(n-1)} + a_{(n-2)} + a_{(n-3)} + 2^{n-3}$$
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1answer
35 views

permutation problem. GMAT-related.

There are 3 elves and 3 dwarves and 6 chairs. The elves and dwarves are trying to integrate with each other and will only sit next to someone of the opposite race and not next to their own kind. How ...
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1answer
43 views

simple combinatorics question - what did I do wrong?

I was asked the following question. I solved it, I thought my solution is correct, but it turns out I was mistaken, I'd like to know why. Question: How many ways are to order 4 sets $(A,B,C,D)$ such ...
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1answer
239 views

A committee of 12 is to be formed from nine women and eight men. In how many ways can this ..

Problem : A committee of 12 is to be formed from nine women and eight men. In how many ways can this be done if at least five women have to be included in a committee ? Solution : Case I : 5 W ...
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2answers
185 views

Calculate product of transpositions

I've searched for this kind of question-answer, but didn't managed to find one because the problem is quite specific. Let me explain: I have permutation: $(13927)(5846)$ which I must write as ...
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1answer
125 views

Probability of all objects not in order

There are four envelopes and corresponding $4$ letters.If the letters are placed in the envelopes at random,what is the probability that all the letters are not placed in the right envelopes? ...
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1answer
54 views

Mapping permutations to an index

I'm trying to calculate the distance-table (#of states as a function of the distance) of the 15-puzzle (has been done before in 2005 on a supercomputer by Korf et al.). There are $16!$ different ...
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227 views

Filling in blanks in a multiplication problem knowing only the set of digits in the product and that 9 divides each factor

The 5774 Ulpaniada (part 2 of it) includes the following question: The following multiplication exercises uses all $9$ digits $1,2,3,\ldots,9$. The digits are encoded by asterisks. We are told ...
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149 views

Permutations versus combinations, order or unordered (problem submitted).

A tourist wants to visit six of America’s ten largest cities. In how many ways can she do this if the order of her visits is (a) important, (b) not important? For part (a), I believe the answer is a ...
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2answers
141 views

How many $7$ digits number can be made?

How many $7$ digits number can be made with $1,2,3,4,5,6,7$ so that they are divisible by $11$? (Repetition is not allowed.) I know the divisibility rule of $11$, so the main problem is counting.
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1answer
40 views

Combinatorics Question - Permutations while fixing cases

Here is the wording of my question: In how many ways can a class with 20 students (12 boys and 8 girls elect a class president, vice president, and secretary if each student is willing to serve in ...
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1answer
52 views

Clarification for this combinations/permutations problem?

I've been going through a list of poker hands and their descriptions, and then attempting to calculate their probabilities by first calculating the number of possible hands for the given hand. I ...
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1answer
304 views

Project Euler #453 confusion

So I decided to give a shot on the #453 project euler problem but there is something that confuses me with the numbers given. I decided to start by calculating the possible arrangements of 4 vertices ...
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1answer
53 views

Permuting “123456728905”

In how many ways can 123456728905be permuted such that, neither two 2's nor two 5's are adjacent to each other ? I'm really confused how to ensure those ...
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1answer
39 views

What is the maximum number of iterations before a sequence is repeated

$A = \{a,b,c,d,e\}$ $B = \{f,g,h\}$ $C = \{i,j\}$ $D = \{0,1,2,3,4,5,6\}$ Suppose a four-tuple is constructed by extracting one element from each set at each successive iteration. The stipulation ...
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2answers
68 views

Describing the pattern in which iterations make two, cyclic sets equal

$A = \{a,b,c,d,e\}$ $B = \{a,b,c\}$ $C = \{0,1,2,3,4,5,6\}$ The first few iterations are as follows: $1.$ $a,a,0$ $2.$ $b,b,1$ $3.$ $c,c,2$ $4.$ $d,a,4$ $5.$ $e,b,5$ $...$ I'm trying to ...
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1answer
51 views

Determine possible combinations/permutations for 5 values which can be empty or not

This related to a computer programming rule I'm working on. I have 5 values: t1 t2 t3 t4 t5 and each of these values can be empty or not. Whether each one is empty or not will change the result that ...
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2answers
57 views

The curriculum's permutation

Consider the word CURRICULUM.How can i find number of ways in which 5 letter words can be formed using the letyers from the word CURRICULUM if each 5 letter word has exactly 3 different letters?My ...
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2answers
81 views

Sign of permutation

Let $p,q \in \mathbb N$. How can I calculate the sign of the permutation $$ \begin{pmatrix} 1 & \dots & p & p+1 & \dots & p+q \\ q+1 & \dots & p+q & 1 & \dots & ...
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355 views

Let $\sigma$ be the $m$-cycle (1 2 $\ldots m)$. Prove that $\sigma ^i$ is also an $m$-cycle if and only if $i$ is relatively prime to $m$

I have consulted some other sources, and think that I have some handle on the basic ideas behind the proof, but am having trouble articulating them. I understand that $\sigma$ sends $a_k$ to $a_{k} + ...
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Very silly permutation question

Okay let me briefly explain my doubt. I'll explain some easy problems,so that you can study easily my mind and you can guess what confusion i might be going through right now. This may be silly.But ...
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1answer
61 views

Proof for the probability that $A[i]<A[j]$ in a random permutation

I'm studying a proof on the expected number of inversions in a permutation of $n$ numbers. The numbers are distinct integers from $1$ to $n$. At some points it takes as given that chosen two indices ...
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1answer
53 views

Elementary Properties of Order

Need help with this question: Prove: Let $x = a_1, a_2...a_n$, and let $y$ be a product of the same factors, permuted cyclically. (That is, $y = a_k, a_{k+1}...a_na_1...a_{k-1}$. Then ...
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1answer
432 views

How many four digit numbers between 1000 and 9000 can be made that are odd and number 1 and 5 cannot be used?

attempt: numbers you can use (0,2,3,4,6,7,8,9) = 8 numbers first position: (2,3,4,6,,8, 7) = 6 numbers last position: (3,7,9) = 3 numbers Rest of the two positions = 6 and 5 numbers left answer: ...
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61 views

How do you make a passwords which are 15 to 24 characters long and have at least one digit?

I know that you can use the complement to find this so one case could be that you have no digits which would be 10^24. I'm not sure where to go from here. I'm not sure what you need to subtract or ...
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103 views

Which is the Weyl group of $U(n)$

Consider the unitary group $U(n)$. How does one compute its Weyl group? Is it the same as the Weyl group of $SU(n)$ since $U(n)\simeq SU(n)\times U(1)$?
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94 views

Order of a permutation using its cycle decomposition

If $A=\{1,2,...,n\}$, $\Omega _A$ is the set of all permutations over $A$, $S_n=(\Omega _A, \circ)$, then for any $\sigma \in \Omega _A$, the order $m$ of $\sigma$ (Smallest $m \in \mathbb{N}$ for ...
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113 views

Why is the parity of a permutation an important concept?

In Pinter's A Book Of Abstract Algebra, the author states that: A number of great theorems of mathematics depend for their proof (at that crucial step when the razor of logic makes its decisive ...
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4answers
60 views

Is there a way to assign a number to a combination without finding and numbering every combination?

Imagine I have 4 letters. Is there some algorithm that produces "abcd" -> 1 "bacd" -> 2 "bcad" -> 3 ... etc without finding and numbering every single combination? My goal is to get a number from 1 to ...
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5answers
483 views

How many permutations of {1,2,…, 9} are there that do not start or end with an even number?

How many permutations of $$1,2,..., 9$$ are there that do not start or end with an even number? This is my attempt Condition 1 [Starts with even] => $$4 * 8!$$ Condition 2 [Ends with even] => $$4 * ...
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2answers
57 views

Couple of Counting (how many ways) questions.

1.If I have a group of 10 seats reserved for people, and there are n=>10 total people, how many ways are there to choose who gets the 10 seats? for ex:If there was a definite number of people lets ...
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generating locally random permutations

I have an intuitive notion of 'local randomness' that I am trying to make precise and understandable, and I am running into a bunch of problems. A quick web search failed to find anything relevant in ...
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Distributing $7$ books to $2$ persons such that each person gets at least $1$ book

In how many ways can $7$ different books be distributed to $2$ persons if each person gets at least $1$ book? I did my calculations and my answer is $126$, but the answer stated is $216$.
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1answer
96 views

How to prove that determinant with permutation symbols

How to prove that $$\varepsilon_{ijk}a_{i\ell}a_{jm}a_{kn} = \det[a]\epsilon_{\ell mn}$$ I'm trying to solve this problem with permutation symbol but i can't solve it Help me,please. Thank you ...
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1answer
73 views

Partitions of n with certain conditions

Let $p$ be prime and $n$ be any integer. Suppose $t=(n^{a_n}, \dots, 2^{a_2}, 1^{a_1}) \vdash n$, (i.e. $t$ is a partition of $n$, where we group repeated integers, so, for example, $2^{a_2}$ means ...
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143 views

Number of ways to order a list of permutations by swapping

I want to solve a problem and I have no idea how or where to begin. I don't even know if it's possible to solve. I tried to find any clues in books about discrete maths but I didn't find anything that ...
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2answers
116 views

What is the probability no slots contain more than two balls given I am trying to sort 5 balls into 6 slots?

I am having a difficult time understanding how to approach this problem. Suppose I have $6$ total slots and $5$ balls. Now, I assign the balls at random to the slots. What is the probability that no ...
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2answers
56 views

Are there any nontrivial ways to factor n-cycles into a product of cycles?

I was reading a proof here about the simplicity of $A_n (n \ge 5)$. It states (and proves) a lemma about 3-cycles: A 3-cycle $(a, b, c)$ may be written as $(a, b, c) = (1, 2, a)^{-1}(1, 2, c)(1, 2, ...
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A conjugation of an operator, which commutes with all permutations, still commutes with all permutations

Assume $v:H^{\otimes m}\to H^{\otimes m}$ is a linear operator on the $n^{\text{th}}$ tensor power of a vector space $H$. For each permutation $p$ on the $m$-element set define the linear operator ...