Questions tagged [permutations]

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

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22 views

Number of distinct $n-l$ digit numbers from $n$ digits some of which are repeated?

How to solve this combinatorics problem? I have $n$ digits, such that $m$ of them are equal and the other $k$ are equal ($n = m+k$). How many distinct $n-l$ digit numbers can I make? (Here $l$ is an ...
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0answers
36 views

Finding a permutation matrix for almost identical permuted matrices

Given $A= \begin{pmatrix} 4 & 2 & 3 & 5\\ 3 & 3 & 1 & 0\\ \end{pmatrix} $ and $B= \begin{pmatrix} 3 + \epsilon_1 & 3 & 1 & 0\\ 4 & 2+ \epsilon_2 & 3 &...
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4answers
40 views

How to calculate n if we have r and nPr? (permutations)

There is this practice problem "How many students are in a class if we can pick 3 delegates in 6545 ways?" I've tried solving this by using the permutations formula but I got left with something that ...
2
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3answers
43 views

The number of ways in which a score of $11$ can be made from a throw by three persons,each throwing a single die once is

What is the number of ways in which a score of $11$ can be made from a throw by three persons, each throwing a single die once? My Attempt: I tried to answer this question by counting the possible ...
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1answer
60 views

Calculating the number of possible paths through grid

We are given with four integers $H$ , $W$ , $A$ , $B$ respectively We have a large rectangular grid with $H$ rows and $W$ columns. Find the number of paths from top left to bottom right moving only ...
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1answer
20 views

Calculating permutations for multiple questions and responses

We are administering a survey and trying to determine how many permutations exist for the combination of question numbers and and responses in order to check our work. Specifically, we have a set ...
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2answers
168 views

Permutation vs Variation - To rank 3 people

I am new to permutation and combination and am looking for guidance in the following example: We have 3 people - A, B, C How many ways are there to arrange them into Rank 1,2,3 Looking at the ...
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2answers
55 views

What group is $S_3\times \mathbb Z_2$ isomorphic to?

The group $S_3\times \mathbb Z_2$ has order $12$. I know four groups of order $12$: $$\mathbb Z_{12},\mathbb Z_{2}\times \mathbb Z_{6},A_4,D_{12}.$$ But it seems that none of them is isomorphic to $...
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1answer
41 views

Find the order of permutation groups

Find the orders of the following permutation subgroups of $S_4$: a) The subgroup generated by $(1,2), (3,4)$ and $(1,3)$. b) The subgroup generated by $(1,2), (3,4)$ and $(1,3)(2,4)$. I ...
2
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1answer
84 views

Number of $3 \times 3 \times 3$ magic cubes

A $3 \times 3 \times 3$ magic cube is a three-dimensional array of the consecutive integers $1$ through $27$, with the special property that the sum along any row, any column, any pillar, or any of ...
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2answers
57 views

Arranging men and women in rectangular table, so that in front of each woman is a man.

A group of $4$ men and $4$ women need to be distributed on two opposite sides of a rectangular table, so that in front of each woman is a man. How many options are there? My attempt: I thought that ...
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1answer
32 views

What is the sum of digits in the unit place of all numbers [closed]

What is the sum of digits in the unit place of all numbers formed using 1,2,3,4,5,6 taken all at a time without repeating any of them?
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0answers
36 views

Term for a permutation group every element of which except of identity has no fixed points

A permutation group on a set $D$ is a group whose elements are functions on $D$ and whose composition is function composition. Is there a term for a permutation group every element of which except of ...
3
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0answers
90 views

The number of transpositions between two permutations [duplicate]

Suppose it is possible, through many steps, to move from the permutation $\pi$ to the permutation $\sigma$ by multiplying, at each step, by a transposition. Knowing that they are not necessarily ...
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1answer
56 views

Finding the number of possible combinations for polynomial coefficients

For example, if I have two numbers $m=\text{max power of each coefficient}$ $n=\text{max sum of the power of the coefficients}$ so for $~m=2~$ and $~n=3~$, the polynomial(which consists of two ...
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1answer
39 views

Numbers with $4$ distinct digits exceeding $6000$ formed from $0,2,3,5,7,8,9$

How many numbers with $4$ distinct digits greater than $6000$ can be formed from $0,2,3,5,7,8,9$ ? i try as follows : $$ \begin{array}{l}{\text { unit digit has } 6 \text { possibilities (except } 0 ...
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2answers
32 views

Find the maximum number of words given the length.

Given that $\alpha \in \mathbb{N}$ and $\Sigma = \{\text{a,b,c}\}$ how can we find the number of different words of length $\alpha$? I can see that the sequence of the words is not important, i.e. it ...
0
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3answers
31 views

7 People born any day of the week

I tried to apply the stars and bars theorem so that I can arrange the $3$ people that are left in the $5$ days between monday and sunday and that gave me $35.$ I know that the total ways to arrange 7 ...
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1answer
33 views

Permutation Of String of length n with A B C

So I have to count the number of words of length N can be made with alphabets A B and C. (repetition allowed) But condition is A's should not be together B'S should not be together. and at-most m C's ...
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1answer
70 views

Conjugation by a transposition for permutations

Concerning the conjugation, I learned that two permutations $ \sigma,\pi\in S_n$ are conjugate if exists $\tau \in S_n $ such that: $\pi=\tau\sigma\tau^{-1}$. Also, these permutations are conjugate ...
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1answer
37 views

Multiply a permutation of cycles by a transposition

I started reading about permutations, cycles, disjoint cycles, decomposition,... etc but I am a bit confused about the multiplication! To be precise, I learned how to multiply two permutations but I ...
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1answer
30 views

I have a question related to permutations [closed]

Prove that the number of permutations of $n(>1)$ different things taking at most $r$ at a time when each thing can be repeated any number of times is equal to $\frac{n(n^r -1)}{(n-1)}$
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3answers
40 views

Round Table Permutations and Combinations Question [closed]

A group of 6 boys and 3 girls sit at a round table of 9 seats for a meal. (i) Find the probability where between any 2 girls, there is exactly 2 boys separating them. (ii) The seats are now numbered ...
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0answers
22 views

Permutations to un-shuffle indistinguishable cards from multiple decks?

An "out-shuffle" on a deck of size $2n$ is defined by the permutation $O$, where $$O = \begin{pmatrix} 0 & 1 & 2 & \cdots & n-1 && n && n+1 && \cdots &&...
2
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1answer
77 views

Multinomial permutation indexing

There are 6!=720 permutations of {1,2,3,4,5,6}. Using Lehmer codes each permutation has a lexicographic index. There are 6!/(3! 3!)=20 permutations of {0,0,0,1,1,1}. Is there a way to index these ...
2
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0answers
33 views

Decompose the irreducible representation of $S_4$ in terms of the irreducible representations of $S_3$

how would one go about doing this? I know the character tables of both groups and that the multiplicity of the irreducible representation $V$ in $W$ is $$\frac{1}{|G|}\sum_{a\in G}\chi_V(a)^*\chi_W(a)...
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0answers
20 views

Making a group of $p$ people with $n$ available nationalities

Making a group of p people using m out of n available nationalities can be one of these two scenarios; $m \le p \le n$ or $m \le n \le p$. Using p,m, and n, how to evaluate the number of ways of ...
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2answers
49 views

In how many ways can the $6$ letters in the list $G,H,I,J,K,L$ be rearranged so that $G$ is the third letter in the list and $H$ is not next to $G$?

The $6$ letters in the list $G,H,I,J,K,L$ are to be rearranged so that $G$ is the third letter in the list and $H$ is not next to $G$. How much such arrangements are possible? I'm guessing the ...
0
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1answer
25 views

Do permutations and their inverses always have the same cycle type?

When talking about permutations such as $(1\ 2 \ 3\ 4\ 5)$ and $(5\ 4\ 3\ 2\ 1\ )$, do permutations and their inverses always have the same cycle type? If so, is there a proof of this and why does it ...
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1answer
34 views

Number of errors of $\sigma$ is the same as minimum transpositions to convert $\sigma$ to the identical permutation

Suppose, we have this general permutation $\sigma$: $$ \sigma = \begin{pmatrix} 1 & 2 & \dots & n\\ \sigma(1) & \sigma(2) & \dots & \sigma(n) \end{pmatrix} $$ How do I ...
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2answers
33 views

A class of 20 students has 55% girls. The top 3 students are felicitated with gold, silver and bronze medals. [closed]

In how many ways would the medals be distributed if there are atleast $2$ girls among top $3$? Will the answer be: $$3!\times\left(\binom {11}{3} + 9\times\binom{11}{2}\right)= 3960$$ or: $$3!\times\...
0
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1answer
47 views

Probability of having increasing even numbers, decreasing odds

I am given a set of 9 numbers from 1 to 9 and I am asked what is the probability to randomly arrange them such that the even numbers are in increasing order while the odd numbers are in decreasing ...
0
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1answer
27 views

Permutation of index and permutation of vector of powers of monomial

I am confused about the following easy stuff, Let $$\mathbf{x} =\begin{bmatrix} x_1 & \cdots & x_n\end{bmatrix}^T .$$ Suppose I have the following monomial $$\mathbf{x}^{\alpha}=x^{\alpha_1}...
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0answers
30 views

The number of transpositions in a representation of a permutation is even for even permutation and odd for odd permutation

This is exercise 9.6b from Section. The Rational Numbers in textbook Analysis I by Amann/Escher. On the symmetric group $\mathrm{S}_n$, define the sign function by $$\operatorname{sign} \sigma := \...
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1answer
22 views

How many possible cases are there

I'm a dentist with biiiig gaps in my math. People have normally 32 teeth. If we say any random tooth or teeth can be extracted, what are the different possible cases (is it called permutations?) Are ...
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1answer
33 views

Number of exam forms to be made for $r \times c$ students

An examination hall contains $r\times c$ chairs. These chairs are arranged in $r$ rows and $c$ columns. Let $P_{i,j}$ be the position of the chair that is in the $i^\text{th}$ row and the $j^\text{th}$...
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2answers
67 views

“Cyclic” and “Circular” permutation - Are they different concepts?

"Cyclic" and "Circular" permutations - are these two different concepts? I have been reading about permutation and encountering them in many places. What are the definitions of them, in simple English ...
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1answer
36 views

Combinatorics: Unique, ordered arrangements of indistinguishable items

I'm sure this is a known problem but it defies a Google search! I'm interested in the number of unique Tic-Tac-Toe boards in which all 9 squares are filled, even if they result in non-sensical games ...
2
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2answers
80 views

Arrangements of the word $ABCDEFGGGG$

If we consider the word $ABCDEFGGGG$. To find the number of arrangments for that word, we just calculate: $\frac{10!}{4!}$. But if now we want to find the total number of arrangements for that word ...
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4answers
82 views

A committee of 8 people consisting of 3 men and 5 women are lining up next to each other for a photograph.

Please help me with the following questions: A committee of 8 people consisting of 3 men and 5 women are lining up next to each other for a photograph. i) In how many different ways can they be ...
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0answers
186 views

How to count the number of good permutations? [duplicate]

We are given a sequence : $[1,2,...N]$. Let's consider all the permutations of the above sequence. Say, $a[1]=1, a[2]=2,\dots,a[N]=N$. Now, a good permutation is defined as the one in which there ...
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0answers
20 views

Can the Binary Icosahedral Group be represented in terms of (small) permutation groups

I know that the icosahedral group, $I$ can be represented as $A_5$, and finite groups can be represented as subgroups of (sometimes large) permutation groups. Being a double covering, the binary ...
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1answer
49 views

Infinite permutation: bijection $\pi:\Bbb N\to\Bbb N$. How to prove 𝜋 is really a bijection?

For example: http://oeis.org/A258746 $a(1) = 1, a(2) = 2, a(3) = 3$. For $n \geq 2$, $m = \lfloor \log_2(n)\rfloor$. If $m$ even, then $a(2\cdot n) = 2\cdot a(n)$ and $a(2\cdot n+1) = 2\cdot a(n)+1$. ...
1
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1answer
21 views

Decomposing a symmetric function into elementary symmetric polynomials. [duplicate]

It is stated that any symmetric function can be expressed in terms of the elementary symmetric polynomials. I am trying to do that for the following generating function: \begin{equation} \prod_{1 \...
-3
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1answer
52 views

Linear algebra questions: True or False [closed]

5 vectors in $\mathbb{R}^6$ are always dependent? If $A$ is singular $n \times n$ matrix, $A^T A$ is also singular? If $P$ is a permutation matrix, then $P$ must be singular? A remedy for the accurate ...
0
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1answer
34 views

Largest value of a third order determinant whose elements are 0 or 1

Find the Largest value of a third order determinant whose elements are 0 or 1. My try: \begin{vmatrix} a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \...
0
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2answers
40 views

The word's length with given alphabet

Consider an alphabet $A = \{a, b, c, d\}$. How many words of length 8 can be created using the alphabet $A$, given the fact that it should contain 3 'a' and 2 'b'?
1
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1answer
35 views

Proof of permutations and cycles [closed]

For a permutation p: X → X, let $p^k$ denote the permutation arising by a k-fold composition of p, i.e. $p^1=p$ and $p^k=p◦p^{k-1}$. Define a relation ≈ on the set X as follows: $i ≈ j$ if and only if ...
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1answer
44 views

${}_nP_r$ versus $n^r$ [closed]

If I had $n=2$ different symbols, how many $2$-symbol ($r=2$) "words" could I create? If I had $n=5$ different symbols, how many $2$-symbol ($r=2$) "words" could I create? Would I use the ...
0
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1answer
37 views

Combinations: summation of combinations equalities

Suppose we have two quantities $$ A = \sum^n_{i=0}C^n_i (X_{n-i}X_{i+1} + X_iX_{n-i+1})\\ B = \sum^{n+1}_{i=0}C^{n+1}_i (X_iX_{n-i+1}), $$ where $C^n_i$ is the combination notation, and $X$ are just ...