# Questions tagged [permutations]

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

8,472 questions
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### Denoting sets (permutations,tuples?),

I'll use and example to explain my question: Example: Let there be two distinct types $a$ and $b$ and each type has equal number of sets. There are two sets of type $a$: $\{1,2\}$ and $\{3,4\}$. ...
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### Find a formula for number of orbits under action of $D_{4}$

We colour each side of a square with $k \geq 1$ colours. Find a formula for the number of orbits under the action of $D_{4}=\{ e , r,r^{2},r^{3},s,sr,sr^{2},sr^{3} \}$ on the set of colours. Now as ...
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### Probability of nth draw being same value

Disclaimer at the beginning: this does relate to a homework problem, but is not the actual homework problem itself. Consider a series of numbers, $1, 2,...,n$. We select one value at a time at ...
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### Permutations of alike objects 3

If I have $1$ jar of $20$ balls with $10$ red, $5$ blue and $5$ yellow, $1$ jar of $30$ balls with $10$ red, $10$ blue and $10$ yellow and $1$ jar of $50$ balls with $15$ red, $20$ blue and $15$ ...
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### Permutations of alike objects 2

If I have $1$ jar of $20$ balls with $10$ red, $5$ blue and $5$ yellow, $1$ jar of $30$ balls with $10$ red, $10$ blue and $10$ yellow and $1$ jar of $50$ balls with $15$ red, $20$ blue and $15$ ...
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### Independence of Records of Permutation

For any permutation $\sigma = (\sigma_1,\dots,\sigma_n)$, call $\sigma_k$ a record if $\sigma_k>\sigma_i$ for $1\le i\le k-1$. Let $\alpha_k$ be the indicator for the event that $\sigma_k$ is a ...
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### How many different arrangements are possible with 3 digits numbers from 1-9 if we have to arrange them in ascending order?

if there are numbers 1,2,3,4,5,6,7,8,9 how many different three digit values we can make such that the digits are in ascending order? since there are numbers from 1-9, I understand that the first ...
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### What would be the number of inequivalent $6$-colourings of the faces of a cube?

Consider the different ways to colour a cube with $6$ given colours such that each face will be given a single colour and all the six colours will be used. Define two such colourings to be equivalent ...
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### What is the difference between these two combinatorics problems?

So the first problem is "In how many ways can we arrange the letters in the word Alabama." and the second questions is "In how many ways can we arrange three Mathematics books, five English ...
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### Permutation and Combination/ Counting

Q. Our student clubs are being particularly active. The aquatics club in particular is in training for an upcoming competition. For all parts of this question, working is required, including ...
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### Permutation of multiple object types with one type kept in a fixed range

We have to place 50 objects of 3 types in a row with 50 places. There are 26 objects of type $a$, 16 objects of type $b$ and 8 objects of type $c$. I am working on a problem where objects of type $b$ ...
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### Number of sortable permutations of a set of natural numbers. [on hold]

Define the permutation of the set $\text{{1,2,3,...n}}$ to be sortable if upon cancelling appropriate term of such permutation remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of ...
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### Counting members of equivalence classes of injective functions

Let $A=\{f:\{1,2,3,4\} \to \{1,2,3,4,5,6,7,8\}, f$ injective$\}$. Let $R$ be a relation on $A$ such that $f_0Rf_1 \iff f_0(1)+f_0(2) = f_1(1)+f_1(2)$. If $h \in A, h(n)=n+2$, what's the cardinality of ...
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### Number of times we have to compose a permutation in order to have exactly k fixed points

Let $f = (1 4 6)(2 7 5 8 10)(3 9)$ in $S_{10}$. Find an integer $n$ such that $f^n$ has exactly $7$ fixed points. I provided the exact numbers, but would welcome a more general solution.
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### Demonstration with Vandermonde's Identity

VANDERMOND’S IDENTITY: Let m, n, and r be nonnegative integers with r not exceeding either m or n. Then $${m+n \choose r} = \sum_{k=0}^{r}{ {{m}\choose{r-k}} { {n} \choose {k} } }$$ The exercise ...
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### Why are these two ways of counting the same combinatorial object not yielding the same result?

In how many ways can we distribute 20 black balls and 2 white balls (indistinguishable modulo color) in 5 numbered glasses such that the fifth one doesn't have more black balls than white balls? ...
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### Permutation in mathematica with one guaranteed element [closed]

I just want to create a Permutation-List of elements {a,b,c,d,...} where each Permutation of three elements needs to contain at least one specific element ...
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### Ways in which circles cut each other [closed]

What will be the number of ways n circles cut each other, for n like 2,3 it is doable but is there any pattern which can be used to build a program or some sort of a series to follow up ?
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### Permutations: Notation that uses composition with exponents?

Notation question. Wiki Compositions of permutations claims that compositions : $$\sigma \bullet \pi$$ is the function that maps any element $x$ of the set to: $$\sigma (\pi (x))$$ and that ...
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### In how many ways can n men and n women sit in n seats serially, so that no 3 women sit consecutively [duplicate]

I tried a lot, but couldn't find a way. One thing which I found interesting is that for n = 7, the maximum women can be W W M W W M W = 5. Hope this might help someone to give an answer.
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### $G$ is an abelian group, and a permutation $T$. Suppose that $x-T(x)\neq y-T(y)$ for all $x\neq y$. Show that $p(x)=x-T(x)$ is also a permutation

Let $G=(A,+)$ be an abelian group, T a permutation on $A$. Assume that $x - T(x) \neq y - T(y)$ for all distinct $x,y \in A$. Show that $p(x)=x - T(x)$ is a permutation on $A$. For that I would show ...
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### Prove: Permutation of a root is another root of a polynomial

I read that Galois group is a permutation of the zeros or roots, this is new to me, so, I have a question. How can I prove, all roots of a polynomial are permutation of one another? in other words, ...
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### Bipartite graphs from permutations

Given are $n\geq 1$ permutations of $abcd$. We construct a bipartite graph $G_{a,b}$ as follows: The $n$ vertices on one side are labeled with the sets containing $a$ and the letters after it in each ...
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### How many are the anagrams of the word MISSISSIPPI in which there are no two consecutive I letters? [closed]

Can anyone give me a complementary approach to this problem? That is, the calculation of the total minus the "unwanted". Answer: 1050
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### How many strings of six lowercase letters from the English alphabet contain the letters *a* and *b* with all letters different?

How many strings of six lowercase letters from the English alphabet contain the letters a and b with all letters different? Answer: 7650720 I saw a resolution that considered the two types of strings ...
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### Counting the permutations of a set in which some elements can be repeated

Suppose I have $n$ cards that I've arbitrarily placed into $p$ piles, possibly of different sizes. The cards are available in $c$ colors, and all cards with the same color are considered identical. ...
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### How many 3-word sentences can be created?

I'm doing this old exam in my course and I stumble upon the question: How many $3$-word sentences can be created from $8$ letters A, $8$ letters B and one of each of the letters C,D,E,F,G when each ...
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### In how many ways can $12$ children occupy the six banks of two seats on a Ferris wheel?

1) In how many ways can $12$ children occupy the six banks of two seats on a Ferris wheel? (consider important the order of two children sitting in a given bank). Resp.: 7680 2) What would be the ...
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### How many are the non-negative integer solutions of $𝑥 + 𝑦 + 𝑤 + 𝑧 = 16$ where $x < y$? [closed]

How many are the non-negative integer solutions of $𝑥 + 𝑦 + 𝑤 + 𝑧 = 16$ where $x < y$? Anyone can explain how to think to approach this type of problem? The answer is 444.
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### How many six-letter words can be formed with the letters of the word ‘policy’ such that the vowels can only occur in even positions?

We have $3$ places for evens and two vowels so selecting $2$ vowels for $3$ positions would be $3P2 =6$ but how can I do the same thing for consonants. Now I've left with $3$ places and $4$ consonants....
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### Galois Theory: Quartic and Klein 4 group

I'm studying Galois Theory on my own and already understand the correspondence between field automorphisms and subgroups of the permutation group. However, I am trying to approach it more from the ...
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### How do I generate permutations consisting of all elements in multiple ordered sets (preferably incorporating element priority)?

I am doing some self study and have come up with a problem I'm not sure how to go about solving. Please correct me if I've used any term incorrectly. Purely mathematical or Python approaches are fine. ...
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### Number of partitions of list [closed]

If I have my_list = [0, 1, 2]. I want to figure out how many possible list partitions there are for a list of length $n$. For the above example, there would be: <...
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### balls and boxes; permutation and combination.

There are $2010$ boxes labeled $B_1, B_2, . . . . B_{2010}$ and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves; ...
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### Count the number of permutations of certain cycles type

Suppose we have $n$ elements, assume there is a permutations over $k$ elements among the $n$ elements so $n-k$ are fixed. Let that the permutation over the k elements is represented by permutation ...