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Questions tagged [permutations]

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

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Number of permutations of four letter word using letters anything between 0-2 times each

The first part of the problem involves calculating the number of four letter words possible to form from five letters A, B, C, D and E. Each letter can be used 0, 1 or 2 times. The second part of the ...
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1answer
16 views

Find the derived subgroup of $A_4$

Find the derived subgroup of $A_4$. Since it is $A_4$, for a permutation $\sigma$ to be in $A_4$, $\sigma$ must have a cycle structure of $2$ cycles. Therefore, $\sigma=(ab)(cd)$. The commutator of ...
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1answer
14 views

Total Permutations: From Letters ABCDEFGH so that E and D are not next to each other.

Here's my approach and wanted to verify if it's correct: If $|U|$ is the number of total possible permutations and $|A^c|$ is the total permutations containing the strings DE and ED. Then $|A|$ the ...
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2answers
21 views

Number of high degree terms in multivariate polynomial expansion

I want to expand the following multivariate polinomial $$\left(\sum_{i=1}^{m} x_i\right)^{n}$$ where $m\geq n$ are both integers. For a fixed integer $k\in\{1,...,m\}$, how to find the number of terms ...
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1answer
34 views

Injective homomorphism whose image is contained in the union of two conjugacy classes

Show that there exists injective homomorphism $\tau : \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \rightarrow S_{27}$ whose image is contained in the union of two conjugacy classes of $S_{27}...
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1answer
38 views

Number of Paths to get from One point to Another

What are the total number of paths that can be taken to get from point A to point B ? Rules :- 1)You can move up ,down ,left , right 2)You CANNOT return to a point that you have been to before ie no ...
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4answers
43 views

How many 4 digit numbers can be formed? [on hold]

Given digits: 0, 1, 2, 4, 5, 7, 8 and 9 1.How many 4-digit numbers can be formed greater than 3000 without repetition? [Here we mean no repetitive digits] My answer is 5*7*6*5 .How many 4-digit ...
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4answers
25 views

Using the digits 1 to 6, how many 3 digit numbers can be formed that are divisible by 3?

I can't seem to wrap my head around this problem.How do I make sure that the digit 0 and digits from 7 to 9 are not included ? Provided- Repetition of digits is allowed.
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1answer
25 views

Permutations of length $n$ in which the first ascent occurs in an even position

A permutation $p=p_1p_2...p_n$ has an ascent in position $i$ if $p_i<p_{i+1}$. Consider every permutation to have an ascent in position $n$. How many permutations of length $n$ are there in which ...
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4answers
50 views

In how many ways can $4$ same oranges and $6$ different apples be distributed to $5$ distinct boxes?

If we have $4$ same oranges and $6$ different apples. In how many ways could we distribute them in $5$ different boxes? I have thought the first part of this problem as saying that the $4$ same ...
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1answer
23 views

Difference between permutation and combination formulas for repetition and not

In the theory of permutations and combinations there are several formulas which include permutations with repetition and without , same for combinations. I know the difference between permutations and ...
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1answer
25 views

What do you call subsets of $G$-sets whose elements are permuted under some action $G$?

Given any group $G$ acting on any set $X$ via some left or right action $\varphi:G\times X\to X$ is there a name for subsets $Y\subseteq X$ with the property that for any $g\in G$ wge have $\{\varphi(...
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21 views

Left and right action and permutation representation

(1) Don't be afraid of the length of the question: Relevant is only Lemma 12; everything else is here only to avoid editing for every question: how did you define it? (2) The Lemma 12 is wrong (...
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18 views

Permuting with Sinkhorn normalization: find $\sigma$ s.t. $\bf{min}\frac{dS}{dp}\rightarrow \bf{\sigma} \times \bf{X'}= \bf{X}$

I have a matrix of samples vs. features and would like to organize or permute the features such that the sum of the first derivative along these features is minimized for all of the samples within the ...
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20 views

Proving that the determinant of permutation matrix is the sign of the permutation [duplicate]

Let's say we have a permutation matrix $P$. I know it equals to the product row-switches elementary matrices -$P=E_1 \dots E_t$ where $E_i$ is a row switch. We also know that $\det(E_i)=-1$, and ...
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Given a random permutation of 1 to N, what is probability to get $N-1$ with a certain strategy?

Given a random permutation of $1$ to $N$, let the sequence be $a_1,a_2,\cdots,a_N$. Erase the first $k$ items, and find out the item (let it be $a_I$) which is first item greater than the fist $k$ ...
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0answers
57 views

How many sequences length $n$, taken from $\{1,2,3,…,k\}$ that the sum of the $n$ elements in the sequence will be divisible by $k$.

I wonder if you can help me with this question I am being dealing with. My line of thinking was this: I know that the sequence is of length $n$, so I divided it into $n$ cells. $$x_1+x_2+x_3+\dotsb ...
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0answers
19 views

Permutation matrix preseves volume of a set in Euclidean space

Suppose I have a set $U$ in $\mathbb{R}^n$. For example, $[3,5]$ in $\mathbb{R}$. We know that the volume (length) in this example is $2$. Suppose I have a permutation matrix $P$, I want to show ...
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1answer
36 views

Show that $sgn (f) = 1$ if and only if there is $h ∈ S_n$ such that $f = h ◦ h$.

I could not show it, I could not define the h. I need help, please.I'm not very good at math. Show that $sgn (f) = 1$ if and only if there is h ∈ $S_n$ such that $f = h ◦ h$. Help me. My proof: -> ...
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4answers
183 views

There are 10 marbles in a bag. $6$ are red and $4$ are blue. You must chose at least 1 red marble. In how many ways can you chose three total marbles.

I thought the answer is $^9C_2$ since the first (red) marble didn't count. You have to pick a red marble which reduces the total count from 10 to 9. The answer is 116 possible ways.
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2answers
25 views

Position of the number $89634$ in sequence of five-digit numbers formed by permuting the digits $3, 4, 6, 8, 9$ [on hold]

If all the possible five-digit numbers that can be formed using the digits $4,3,8,6$ and $9$ without repetition are arranged in ascending order, then the position of the number $89634$ is?
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Zeroes of a function with logarithms and peculiar symmetry

I encountered a strange function, defined on an open three-dimensional unit cube centered on the origin and taking values in $\mathbb{R}$. The function has some symmetry properties. It is not ...
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Is there an established name for the system of all orbit of all subgroups $H\le G$?

Is there an established name for the system of all orbit of all subgroups of transitive permutation group $G$? Or, at least, was this or equivalent concept examined in a some book/article? For ...
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2answers
48 views

Proving greedy algorithm for largest permutation with at most K swaps

Let's see the following statement: Given an array with the first N natural numbers in any order perform at most K(non-negative) swaps in order to obtain the largest possible permutation. For making ...
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3answers
62 views

How many $4$-digit numbers can be formed using digits $0,1,…6$ such that it contains the digits $3$ and $5$?

How many $4$-digit numbers can be formed using digits $0,1,...6$ such that it contains the digits $3$ and $5$? My try: All possible $4$-digit numbers $= 7 \cdot 7 \cdot 7 \cdot 6$ $4$-digits ...
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0answers
24 views

Optimization of sum of squares over permutations

Suppose I have fixed, positive values $n_1, \cdots, n_\ell$ and $T$. I'm looking for an algorithm to optimize \begin{align*} f(\boldsymbol{n}) = T\left(\sum_{j=1}^{\ell}\left(\sum_{i=1}^{j}n_i\right)^...
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1answer
51 views

The game of balls

$6$ balls marked as $1,2,3,4,5$ and $6$ are kept in a box. Two players A and B start to take out $1$ ball at a time from the box one after another without replacing the ball till the game is over. The ...
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3answers
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Combinatorics: Distinguishable and Indistinguishable Variables

$3$ men and $5$ women (each of the $8$ being different from all the rest) are lined up for a photograph. Also in the line are $3$ identical armadillos which are completely indistinguishable from each ...
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1answer
42 views

How many numbers are there between $100$ and $1000$ such that every digit is either $2$ or $9$?

How many numbers are there between $100$ and $1000$ such that every digit is either $2$ or $9$? I couldn't understand what actually the question means. Does it talk about the numbers like $222$, $...
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All factors out of the total obtained which are multiple of 5 is [closed]

All possible tow factors products are formed from the numbers 1, 2, 3, 4, 5, ... 200. The number of factors out of the total obtained which are multiple of 5 is.
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1answer
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Number of arrangements of $5$ boys and $4$ girls such that all $4$ girls do not come together

There are $9$ students of which $5$ are boys and $4$ are girls and we have to find the number of arrangements of all the $9$ students in such a way that all the $4$ girls do not come together. I came ...
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Circular arrangement, probability of winning a game. [duplicate]

A host and 9 guests are seated at a circular table. Your host is feeling generous: she places a gold coin in front of her, and announces that one of you will be taking it home. Whoever has the coin —...
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1answer
39 views

The normalizer subgroup of $\{(),(12)(34)\}$

Let $H$ be the subgroup $\{(),(12)(34)\}$ of $S_4$. Find the normalizer $N(H)$ and the quotient $N(H)$. What I have so far: in this case, $N(H)=Z((12)(34))$. The order of the conjugacy class of $(12)(...
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When do two permutations commute?

Let $\sigma$ $\in$ $S_n(Symmetric\;Group)$ .Let $S_\sigma$ := [i $\in$ {1,2,....,n} | $\sigma(i)=i$] .I have to show that if $\phi$ and $\tau$ are two disjoint cycles, then $\phi^i$ and $\tau^j$ are ...
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1answer
36 views

How do I solve this problem using Permutation and Combination? [closed]

The sum of proper divisors of 72 (1 and 72 excluded) is i. 195 ii.122 iii.194 iv. None of these I have already solved it by adding the divisors (which was easy to do and the only approach I could ...
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2answers
60 views

Number of ways to give chocolates.

There are $N$ students and $M$ types of chocolates. How many ways are there to distribute the chocolates to the students given that, no more than two consecutive numbered students, gets the same type ...
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1answer
21 views

Permutation ( selection of 4 letters from 12)

Question : how many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no R's and two E's? My attempt: If there are to be no R's then the selection is ...
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2answers
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Seating arrangement ( permutation)

So I tried this question but somehow I have a hard time understanding what they ask . The question goes : calculate the number of ways 3 girls and 4 boys can be seated in a row of seven(7) chairs if ...
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2answers
28 views

How many different $7$-digit numbers can be formed from $0,1,2,2,3,3,3$ assuming no number can start with $0$? How many numbers will end with $0$? [closed]

I got this question and wanted to confirm my solution. How many different $7$-digit numbers can be formed from $0,1,2,2,3,3,3$ assuming no number can start with $0$? How many numbers will end with ...
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How to solve $\beta^{-1}\pi^{2016}\beta = \alpha$ where $\alpha$, $\beta$ are given permutations

I am given two permutations $$\alpha = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 5 & 6 & 7 & 8 & 3 & 11 & ...
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1answer
28 views

Number of permutations of $3$ t-shirts out of $4$

Mr. A has a set of $4$ distinct t-shirts. Since it is winter he has to wear $3$ t-shirts everyday to beat the cold. How many distinct arrangements of t-shirts can he wear anyday? (Here is he has t-...
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2answers
66 views

How many $10$-digit numbers (allowing initial digit to be zero) in which only $5$ of the $10$ possible digits are represented?

The answer I found was $$(5^{10}-|\text{only}~4~\text{digits}|-|\text{only}~ 3|-|\text{only}~ 2|-|\text{only}~1|) \cdot C(10,5)=$$ where $|\text{only}~1~\text{digit}| = 1^{10} \cdot C(10,1)$ $|\...
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3answers
35 views

Remove and replace letters in 'Arkansas'

A sign reads 'Arkansas'. Three letters are removed and put back into the three empty spaces at random. What is the probability that the sign still reads 'Arkansas'? My method I tried a different case ...
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1answer
22 views

For the following permutation in $S_9$, compute the sign, order and give cycle decomposition into disjoint cycles

For the following permutation in $S_9$, compute the sign, order and give cycle decomposition into disjoint cycles, justifying your steps. $\sigma \in S_9$ with $$\sigma(1)=6, \sigma(2)=4, \sigma(3)=...
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Special subsets of Symmetric group $S_n$ from Barrington's theorem

With inspirations from the proof of Barrington's Theorem, I have the following questions about symmetric groups $S_n \; (n \ge 2)$. Enumerate all proper subsets $T$ of $S_n$ satisfying that $\; \...
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1answer
22 views

Counting permutations with additional requirements

Say I have a set of 36 objects 0-9A-Z. I want to find out how many different orderings of a dozen selections I can make. This is just 36P12, which is a large number. However, I have two scenarios ...
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1answer
45 views

Number of ways to arrange $a,a,b,b,c,d$ in the grid such that no row is empty

Find Number of ways to arrange $a,a,b,b,c,d$ in the grid below such that no row is empty I tried to use principle of inclusion exclusion taking total possible minus empty rows. that is we have ...
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4answers
66 views

Notation of symmetric sum notation

When you use the symmetric sum notation, for example, $$\sum_\text{sym}abc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times? I am confused about repetitions of the same ...
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2answers
58 views

Number of ways to divide $20$ distinct objects into five groups of size $6,6,6,1,1$

Number of ways to divide $20$ distinct objects into five groups of size $6,6,6,1,1$ For this logically the answer should be $$\frac{20!}{(6!)^3} \times \frac{5!}{3!2!}$$ Since the quintuple $6,6,6,...
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1answer
73 views

Counting and Finding Simple Permutation Groups within Symmetric Group Sn

Is there any explicit mathematical result or algorithm about the total number of simple permutation groups within a Symmetric Group Sn? and what are them? Thanks.