Questions tagged [permutations]
For questions related to permutations, which can be viewed as re-ordering a collection of objects.
9,725
questions
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Represent a permutation as a product of disjoint permutation cycles.
$\pi_1=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 1 & 4 & 2}$ and $\pi_2=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 2 & 3}$. Let $\pi_3$ ...
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23 views
Count the number of permutations in a decreasing space
Let me start off by saying that I'm not a mathematician, so this is probably an easy problem to solve, but I haven't been able to yet..
The problem is that I want to place $n$ objects on a grid with $...
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1answer
30 views
Please help me with the logic behind this probabilty question. Three numbers are chosen at random without replacement from (1, 2, 3 …, 10).
Question: Three numbers are chosen at random without replacement from
$(1, 2, 3 \dots, 10)$. The probability that the minimum number is $3$
or the maximum number is $7$.
My logic: since the ...
0
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1answer
45 views
Homework Help: Probability of 5 element subset having one prime and a single digit
Here's the question:
Determine the probability that a randomly chosen 5-element subset of numbers from 1 to 20 contains at least one single digit number and at least one prime number.
Hi. Currently ...
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1answer
25 views
How many ways can I arrange $x$ zeroes in $n$ spaces [closed]
I cant figure out the formula for arranging $x$ zeroes in $n$ spaces as shown below
...
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0answers
11 views
Deriving the formula of number of geometrical isomers in symmetric polyenes [duplicate]
For example, if we take the case of Hexa-2.4-diene we have 3 isomers which will have cis-cis, trans-trans and cis-trans arrangements at the two double bonds present. The aim is to get a formula for ...
2
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1answer
81 views
Isomorphism between $S_n$ and a subgroup of $S_{n+1}$
Let $S_n$ be the symmetric group on $n$ letters. Now, I wish to show that $S_n$ is isomorphic to the subgroup of all the elements of $S_{n+1}$ that leaves $n+1$ fixed.
Let $\sigma \in S_n$. Define ...
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3answers
38 views
Distributing balls into distinct boxes
So basically my question is this:
I have $30$ non distinct balls that I want to put inside $4$ distinct boxes. For every $1\le i\le 4$, the $"i"$ indexed box must at least have $i$ balls and at max $...
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0answers
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Deriving the general formula of $\mathcal{\epsilon_{ijk}} \mathcal{\epsilon^{ijk}}$
As stated in the title, with $i,j,k=1,...,N$.
I expanded $\mathcal{\epsilon}_{ijk}\mathcal{\epsilon}^{ijk}$ as follows:
$$\mathcal{\epsilon}_{ijk}\mathcal{\epsilon}^{ijk}=\underbrace{\mathcal{\...
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0answers
18 views
Why does this happen while distributing cards and what did I do wrong?
If I have $52$ cards that are to be distributed among $4$ players in sets of $17, 17, 17, 1$, in how many ways can I distribute the cards?
I approached the problem in two ways:
I figure out the ...
0
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1answer
24 views
How many different passwords of length 4 can be made with atleast 1 digit in it.
Consider the character set of total 62 characters : - 26 capital letters 26 small letters 10 digits
I know that i can solve this by
Total passwords - Total passwords only containing characters
...
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1answer
19 views
how many password of length 5 can be made with exactly two digits in it. [closed]
Consider the character set of total 62 characters : -
26 capital letters
26 small letters
10 digits
10 x 10 x 52 x 52 x 52
is this answer correct? If not please explain in detail different ways of ...
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0answers
24 views
How many ways are there to stack m piles with n items?
How the calculate the number of ways (permutation) of stacking m piles with n items?
Let's say I want 2 piles with 3 items: How to caluculates the permutations? A pile can consist of no item. What's ...
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0answers
14 views
Is there a simple reduction from permutations of {1, … 2M} to {1, … M}?
Suppose I have a random permutation uniformly chosen from the set of all permutations of $M N$ elements; often in the contexts that I am interested in, this is going to be $2^{m+n}$ elements. And let ...
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2answers
65 views
How many solutions are there for $x_1+x_2+x_3+x_4 = 49$? [closed]
How many solutions are there for this equation $$x_1+x_2+x_3+x_4 = 49\,,$$ where $x_i,\;i= 1,2,3,4$ is a non negative integer such that:$$ 1\le x_1\le 8,\;3\le x_2\le 9,\;10\le x_3\le 20,\ 0\le x_4\,?$...
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0answers
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combination problem with addition [closed]
Please tell me what are the ways getting 37 by adding 6 digit using 1 to 17.No restrictions.
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3answers
59 views
40 people are around a table,how many ways we can choose 5 person in case that between every 2 chosen person should be at least 3 people
$40$ people are sitting around a table. In how many ways we can choose $5$ person of them so that between every $2$ persons that are chosen will be at least $3$ other people sited.
I think it is ...
0
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1answer
16 views
how many ways are there to share 12 white ball and 3 red ball between 5 person in case that… [closed]
How many ways are there to share 12 white balls and 3 red balls between 5 person in case that :
a) every person should have at least one ball and non of them should receive more than one red ball.
b)...
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1answer
22 views
combinatoricsd problem [closed]
In how many ways, 10 girls and 5 boys can be arranged in a row; So that no two boys are together?
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0answers
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cows perms and combs q - tricky! [closed]
13 cows are placed in a 16 yards by 9 yards fields. Prove that there must be at least 2 cows which cannot be separated by more than 5 yards apart
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37 views
How would you find the number of permutations of n? [closed]
How would you find the number of permutations of $n$, where
$n=a+2b+3c$ and $b \leq 3$ all are nonnegative also
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0answers
9 views
Variation of Counting ways to fill a 3×3 grid
Let every cell of $3 Ć 3$ array is filled by natural number such that $x_1*x_2*x_3 = y_1*y_2*y_3 = 2^3 * 3^4 * 5^7$
where $x_i$ , $y_i$ are product of numbers filled in three cells of $i$ th row and $...
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0answers
8 views
Normal closure of the Weyl subgroups [closed]
What is the normal closure of the Weyl subgroup in the general linear group GL(n,k)?
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0answers
34 views
A precise formula for the summation of an inner product
We have 2 strings $|v\rangle$ and $\langle u|$,
$|v\rangle=|e_{1}\rangle^{np}|e_{2}\rangle^{n(1-p)}$ where $e_{1}$ occurs $np$ times and $e_{2}$ occurs $n(1-p)$ times and $\langle u|=\langle f_{1}|^{...
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1answer
16 views
Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions.
Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions as follows:
$$
(a_1 ... a_{k-1} a_{k})=(a_1 a_{k})(a_1 a_{k-1})...(a_1 a_2).$$
I found an answer here: ...
0
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1answer
24 views
is this the correct way to solve the question?
There are 35 students in David's homeroom class. There are 5 students who take English and Biology,and 7 students who take neither of these subjects.There are 3 more students taking English only than ...
1
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1answer
39 views
Let us subdivide $n$ persons into $m$ teams. How often we must rebuild the teams that everyone worked at least once with everyone else in a team?
We need to subdivide $n$ persons into $m$ teams of equal size ($m\mid n$). How often we must rebuild the teams (by shuffling the team members / reassigning them to a different team) so that everyone ...
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0answers
13 views
Probabilites with non repeating numbers
I am trying to calculate the likelihood of picking x numbers from a selection out of y whe order is important.
I have the following for working out the when the numbers can be repeated, for example ...
1
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1answer
25 views
Arrangements and permutation
How many arrangements of $A, B, C, D, E, F, G, H$ are there such that $A$
and B are between C and D ?
Attempt : I am trying to solve this problem using simple counting process like first place can ...
0
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1answer
51 views
How many possible possible passwords are there for 8-100 characters?
Requirements/Restrictions: Minimum of $8$. Maximum of $100$. At Least $1$ letter from the latin alphabet (capitalisation doesn't matter-g is same as G, $26$ letters), atleast $1$ number ($0-9$, $10$ ...
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0answers
89 views
Stabilizer and orbit of a polynomial.
I was reading permutation part where I found this problem.Let there be a polynomial group $S_4$ where$p=x^2_1x_2+x_2^2x_3+x_3^2x_4+x_4^2x_1$ with variable $x_1,x_2,x_3,x_4$. Find the stabilizer and ...
1
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1answer
24 views
Combinatorics - how many ways to divide balls in two groups
Suppose I have:
8 black balls
3 white balls
5 blue balls
how many ways there are to divide those balls into two different groups (note that there is no need to divide into two groups with even ...
-1
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1answer
43 views
Closed form of $\sum_{n=1}^{\infty} (n-1)!x^n$ [duplicate]
In a problem that I am trying to solve using generating function, the right-hand side (RHS) of the generating function equation is
$\sum_{n=1}^{\infty} (n-1)!x^n$.
Would like to find the closed ...
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0answers
21 views
How many arrangements are there of the word āPERMUTATIONSā so that no three vowels come together?
I did this-
Total Arrangements without any constraint - Total arrangements when 3 Vowels together = Total arrangements when no 3 vowels are together.
$\frac{12!}{2} - (\frac{10!}{2} \cdot {}_5C_3 \...
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1answer
42 views
password security combination problem
Been trying to do this question since 3 days yet not been able to approach please!!! help
If we disable and technically block all easily guessable weak passwords (specific combinations of letters, ...
2
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0answers
54 views
Circle permutation without consecutive integers placed together
How many ways can $n$ numbers from 1 to $n$ be arranged in a circular order without consecutive integers being placed together? (Note: 1 and $n$ are also considered consecutive integers)
For example, ...
2
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3answers
51 views
Cards Shuffle problem: How to prove solution exists? Is there a formula for the solution?
Problem
You have a pile of N cards sorted from 1 to N where card 1 is the one on top and card N is the one at the bottom.
We do a shuffle operation on the N cards and the shuffle consists of going ...
0
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1answer
28 views
Easy question on combination
In how many combinations can I arrange 10 different pens in 5 glasses?
I found this problem but I'm not sure of what exactly is asking.
Is it correct to interpret it as a simple combination, namely:
$...
0
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2answers
27 views
Geometry and Permutation and Combination problem
In an n sided regular polygon, the formula for the number of diagonals is $\frac{n(n-3)}{2}$. This formula is derived by selecting $2$ out of $n$ sides i.e. $n\choose 2$ and then subtracting the ...
3
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0answers
44 views
About probability of arranging books on a shelf
Lets suppose that we have 4 math books, 6 statistics books, and 10 books of other subjects, and we want to order a shelf with those books. The restriction we have is that the math and the statistic ...
2
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1answer
38 views
How do I show two pairs of elements of $S_n$ are conjugate by the same element?
Let $\alpha, \alphaā, \beta, \betaā$ be distinct non-identity elements of $S_n$. Suppose there exists $\tau \in S_n$ such that $\alphaā = \tau \alpha \tau^{-1}$ and $\betaā = \tau \beta \tau^{-1}$.
...
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2answers
49 views
Why are there no cycles in this “1234-graph” on the permutations
Fix $n\in\mathbb N$, and consider the following directed graph whose vertices are the permutations of length $n$:
For a permutation $p=[p_1,..., p_j, ..., p_k, .., .p_n]$ with if $p_j+j=k$, then ...
2
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1answer
43 views
Finding the no of V-shaped Permutations?
Consider $n$ distinct real numbers: $a_{1}, a_{2}, \ldots, a_{n},$ A permutation $([1],[2], \ldots,[n])$ of the indices $\{1,2, \ldots, n\}$ is said to $V$ -shaped if there exists an integer $r \quad(...
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0answers
37 views
is there any way to find the maximal length of cycle in a permutation group?
How we can find the biggest cycle of a permutation group with given the set of generators ? is there any algorithm or theorem for that?
Or how we can find the set of generators with biggest cycle as ...
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0answers
39 views
A permutation Ļ on $[n]$ is said to be even-dominated if $\phi_{2iā1}< \phi_{2i}> \phi_{2i+1} \ for \ all 1 ā¤ i < n/2 $
Let a be the number of even-dominated permutations on $[n]$. Let $a(x)$ be the exponential generating
the function of $(a_n)_{nā„0}$.
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0answers
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How do I show this combinatorics statement is true? [closed]
This is what I have.
$\frac{n!}{(n-r+1)!r!}$ + $\frac{n!}{(n-r)!r!}$
-> $\frac{n!(n-r)!+n!(n-r+1)!}{(n-r+1)!(n-r)!r!}$
-> $\frac{n![(n-r)!+(n-r+1)!]}{(n-r+1)!(n-r)!r!}$
I believe I want to get
$\...
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0answers
20 views
A question about permutation with same elements/repetitions
My math textbook states that the formula for the permutation of $n$ elements where $r$ elements are selected is, $$_nP_r\text{ (with repeating elements)}=\frac{_nP_r}{x_1!\times x_2!\times\cdots\times ...
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0answers
8 views
Integral solutions of a linear equation with constraints
For linear equation like:
$$x_1+ x_2+ x_3+ \dots x_r = n$$
All non negative integral solutions are: $\binom{n+r-1}{r-1}$
CASE I: If constraint is given like $x_1\geq a, x_2\geq b,\dots$
We can ...
0
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1answer
37 views
Is this the correct way to solve the question below
Five character security codes for a store,are created by using a letter followed by 4 non-repeating numerical digits or using a digit followed by 4 non-repeating letters.
The total number of security ...
0
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1answer
21 views
How many ways can we select 14 numbers if we want to use each even digit at least once and the numbers can be repeated?
How many ways can we select 14 numbers if we want to use each even digit at least once and the numbers can be repeated?
I figured that, we should take the maximum number of possibilites and extract ...
