# Questions tagged [permutations]

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

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### Confusion over Combinations and Permutations

Just when I thought I understood everything, I have yet again made myself confused and cannot resolve this issue. Consider selecting 3 people from 5 where the order of selection matters, this is ...
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### Does the Symmetric Group on 71 letters get specific attention by researchers in simple groups? [closed]

Since it contains all the sporadic groups and no smaller symmetric group does, it might be of special importance.
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### Selecting subsets $G$ of a set $\mathcal{K}$ of integers so that atleast one subset of $G$ has consecutive integers in wrap around sense.

Let $\mathcal{K}=[1,2,\cdots,K]$ be a set of cardinality $K$. For parameters $a$ and $p$, taking integer values, how many subsets $G$ of $\mathcal{K}$ exist of cardinality $(1+a+p)$ such that there is ...
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### randomized left stochastic matrices: two by two right multiplied by two by one as an average

I'm looking at the identity matrix and its inverse right multiplied by two by one matrix [1,0] and two by one matrix [0,1]. Choosing either is a fifty fifty chance. If I simply right multiply and ...
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1 vote
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### Number of ways to arrange characters in the alphabet [closed]

Given an alphabet of size A and a bowl of alphabet soup of size S. Assuming that the distribution of characters in any one bowl is uniform, what is the likelihood that a random bowl contains at least ...
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### Exchangeability in terms of permutations in card theory, Ethier (2010).

Exchangeability in terms of permutations in card theory, Ethier (2010). I am experiencing doubt with reconciling the standard definition of exchangeability with the way Ethier (2010), in Doctrine of ...
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### In how many different ways can we color 12 balls into red, green and yellow? [closed]

So basically, I'm stuck between two methods, and I don't know which one is correct. Am I supposed to solve it like this 12!/3!9! = 1211109!/3!*9! = 1320/6 = 220 differnt ways or. Am I stupid and its ...
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### What is the probability that one simultaneous roll of five dice gets a four? [closed]

When playing Yahtzee!, five dice are rolled simultaneously. What is the probability that one roll of five dice gets a four. That is that four of the dice each show k st dots at the same time as a die ...
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### Transposition $q(j)$ equal to permutation $p(i)$? [closed]

The book tells me that, for any permutation $p$ and a transposition $q=p.t$, $q(i)=p(i)$ Now, the basic transposition is expressed as: $$\sigma \begin{pmatrix} j&i\\\\ i&j \end{pmatrix}$$ So,...
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### How to find different representation of permutation in product of transpositions?

It is known that product of transposition to write is not unique. it can be written in many ways,there is theorem that stats that all representation of product of transposition for a given cycle are ...
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### Computation of the symmetric number of a finite group [duplicate]

I got a bit curious about the concept of the symmetric number of a finite group, and decided to do some computations with GAP to determine their values for some small finite groups. The symmetric ...
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### Maximizing the number of colors so that every subgrid contains all colors

Consider an $n\times n$ grid. Define the set $S$ as subgrids shapes which includes all $(i,j)$ pairs so that $i\times j=n$. eg: we can take $i=1, j=n$ which is a row shape structure and it belongs to ...
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### Number of arrangements of "AABCXX" without "AA" and "C" isn't next to "A" or "B".

I'm trying to use the Goulden-Jackson cluster method with the alphabet $V=\{A,B,C,X\}$ and forbidden bad words $\{ AA, AC, CA, BC, CB\}$. I also want to keep track of how many of each letter there are,...
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### A Natural Probability Distribution on the Infinite Symmetric Group

Is there a "natural" probability distribution on the set of bijections from $\mathbb{N}$ to itself? Preferably, I would want a distribution which arises from some combinatorial procedure. ...
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### Permutations Doubt: How many $5$-digit whole numbers with no $0$s are divisible by $6$?

I came across this combinatorics question that was quite interesting, as in methods to solve this. How many $5$-digit whole numbers with no $0$s are divisible by $6$? First, I found the number of $5$-...
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### Permutations of 10 players within 2 Badminton courts: Covering $10$-vertex complete graph $K_{10}$ by two disjoint $K_4$

I am facing this everyday problem and I wanted to actually see how to formalise and reason on. We have 10 players and two courts in our badminton matches. We define a shift to be an instance of ...
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### In a tournament, there are twelve players $S_1,S_2,....,S_{12}$ and divided into six pairs at random

In a tournament, there are twelve players $S_1,S_2,....,S_{12}$ and divided into six pairs at random. From each game a winner is decided on the basis of a game played between the two players of the ...
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### Can we do any better bijective mapping of a permutation series which is only bijective for a probabilistic subset of it's input domain

So we want to bijectively map one path to another. But depending on start and target node we can only choose from a subset of all transitions. It would look like this: We also do not know where one ...
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### permutation of the determinant according to the groups $S_{n1}$ and $S_{n2}$?

Reading about alternating linear n maps, I found this alternative definition of a determinant, based on its permutation expression (which iteratively sums the product of all the permutations that can ...
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### Understanding the generalized "Birthday Problem" formula [duplicate]

While practicing frequently asked probability questions during interviews, I came across the classical "Birthday problem". While I understand some of the reasoning explained on wikipedia, ...
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### Combinatorics question related to stars andbars #permutation #combination [closed]

A community with n members chooses its representative by voting. a) In how many ways can “open” voting result, if everybody votes for one per- son (perhaps for himself/herself)? Open voting means that ...
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### Number of ways in which 5 girls and 5 boys can be arranged in a line such that only 4 girls stand adjacent to one another

I have already calculated n but i am confused to find the value of m. I have tried the following process:- Selected 4 girls from 5 girls in ${}_5C_1$ ways and treated it as a unit After selecting 4 ...
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### combinations problem - teachers to groups [closed]

My question is in how many ways can you divide 20 teachers into two groups, one group would have 15 people and the other would have 5. I tried Benjamin Dickman's formula but it did not work out =, the ...
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### Number of all possible arrangements such that no two objects of same kind are together

There are $k_1+k_2+k_3+...+k_n$ objects of which $k_1$ are of first kind, $k_2$ are of second kind, $k_3$ are of third kind, ..., $k_n$ are of $n^{th}$ kind. Calculate the number of all possible ...
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1 vote
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### How to find the number of elements in $S_7$ that commutes with $(123) (245) (456)$?

The following question was asked in my masters entrance exam and I would need help in finding the correct number. Question: The number of elements in $S_7$ that commutes with $(123) (245)(456)$ is ......
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### Most efficient way to recompute matrix cost after swapping columns

I work with regular matrices $M \in \mathbb N^{n\times n}$ with $n > 1$. My cost function is: $$\text{Cost} = \sum_{i=1, j>i}^n M_{ij}$$ Hence, we only consider the superior triangle, diagonal ...
1 vote
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### Counting $10$ length paths in a $2 \times 4$ rectangle with distance $6$ units from start to end meaning negative moves allowed?

How many different routes of length 10 units (each side is 1 unit) are there to traverse from lower left corner (point A) to top right corner (point B) in a rectangle with 2 rows and 4 column cells ...
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### I'm trying to create a match schedule for pickleball

I have two teams each with 12 members. Matches will be doubles play. I want each member to play only once with each teammate. That's a total of 11 games each, 66 in total. I want each player to play ...
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### Formalising the problem and create a proof for the game "Waffle"

Waffle is an online game at https://wafflegame.net/daily. It consists in moving letters (swapping them) to recreate the original words. While you have 15 moves, it can be done in 10. I usually try to ...
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### How to determine what is n and what is r?

In case of permutation with repetition, we have formula = $n^r$ How do you decide which thing will be $n$ and which will be $r$? Like in this question: Your mother-in-law buys 1000 small gifts to give ...
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### Combinations and forming groups [closed]

In how many ways can three groups of 3 be made from letters of the word CROCODILE? PS: Answer is 140 but I fail to see the logic. My take was (9C3 *6C3 *3C3) / (3! *2! *2!) =70 but its wrong
### There is a permutation matrix $P$ such that $PAP^{T}$ is in this form for symmetric $A$
Suppose that $A$ is a real matrix and is symmetric and nonzero, then I want to prove that there is a permutation matrix $P$ such that \$PAP^{T}=\left[\begin{array}{ll} B & E^{\top} \\ E & C \...