# 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 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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### “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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### 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 ...
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### 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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### 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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### 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, $a=1, a=2,\dots,a[N]=N$. Now, a good permutation is defined as the one in which there ...
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### 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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### 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 $a(2\cdot n+1) = 2\cdot a(n)+1$. ...
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### 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 \...
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### 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 ...
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### 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} \...
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### 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'?
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### 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 ...
### ${}_nP_r$ 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 ...
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 ...