# Tagged Questions

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

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### Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
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### Probability of drawing $m$ of $A$ in $n$ cards given a deck of $d$ cards contain $a$ copies of $A$?

As in the title I'm trying to work out what the chances of drawing $m$ copies of a specific card in $n$ draws are given a deck size of $d$ containing $a$ copies of $A$. I've tried using permutations ...
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### Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
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### Multi-stage Probability

I think the easiest way to explain what I'm having trouble with is to give an example question: A monkey is given 12 blocks: 3 Squares, 3 Rectangles, 3 Triangles, 3 Circles. Calculate the probability ...
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### find all combination without overlapping

(*Constraint) Kinds of number should be limited in 3 (ex. {1,1,1,1} o/ {1,2,3} o / {1,1,2,2,5,5,4} x) And I want to find series of Integer. For example, if n = 4 (n is length of numbers) {2, 7, 2, ...
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### Find the number of seating arrangements at a round table of three single men, two single women, and two families

Three single men, two single women and two families take their places at a round table. Each of the two families consists of two parent and one child. Find the number of possible seating arrangements. ...
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### In how many ways can a committee of $6$ people be selected from $7$ men and $6$ women if it can contain at most one of persons A and B?

A committee of $6$ people will be formed with $7$ men and $6$ women. The oldest of the $7$ men is A and the oldest of the $6$ women is B. It is described that the committee can include at most one of ...
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### Number of permutations of $S_n$ such that $\sigma^h(a) = \sigma^k(b)$

A basic result in combinatorics is: In $S_n$ there are $(n-d)(n-2)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a \neq b$; $d(n-1)!$ permutations $\sigma$ such that ...
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### A committee of 12 is to be formed from nine women and eight men. In how many ways can this ..

Problem : A committee of 12 is to be formed from nine women and eight men. In how many ways can this be done if at least five women have to be included in a committee ? Solution : Case I : 5 W ...
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### What is the usage of Permuation and Combination in the real life or Computer Programming world?

I have never understood the application of P & C, even in the software programming. I know, the principle behind P&C is counting. So, basically it makes the counting quicker. However, I could ...
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### How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
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### Is there any algorithm to find Isomorphism function between two graphs?

Two graphs which contain the same number of graph vertices connected in the same way are said to be isomorphic. Formally, two graphs and with graph vertices are said to be isomorphic if there is a ...
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### Probability; bridge hand question

$13$ cards are chosen at random with no replacement from a deck of $52$ cards. find the probability there are $5$ spades chosen, $4$ hearts, $3$ diamonds and $1$ club. I got ...
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### Properties of lists with arbitrary lengths and alphabet size

I am having trouble understanding a problem of applying the concepts of permutations and combinations in an example that I found while reading my textbook. Basically, it wants the number of elements ...
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### How many different varieties of pizza can be made if you have the following choices:

How many different varieties of pizza can be made if you have the following choice: small medium, large; thin, hand tossed, pan; and $12$ toppings (cheese is an automatic), from which you may select ...
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### How can I check whether the group $[16,13]$ in GAP with $3$ generators can be generated by $2$ elements?

The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same ...
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### What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is …

What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is $2^33^3$ where $x, y,z\in \Bbb N$? What I tried : At least one of $x, y$ and $z$ should have factor ...
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### Prove that (01) and (01…n-1) generate Sn? [duplicate]

Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1). I understand that this can be solved using induction, and I've set up my base cases. ...
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### G-set of 8 colors and 6 sides

"How many distinguishable wooden cubes can be painted if u use 8 colors (different colors on every side)" I have solved this question using Burnside's lemma ...
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### Fraction of permutations satisfying a poset

Let $[n]:=\{1, ..., n\}$. Let $P$ be a poset on $[n]$. What is the fraction of permutations that satisfy $P$ when we view a permutation as inducing a linear ordering on the numbers? For instance, if ...
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### In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
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### Find the number of 3 letter words that can be formed from the word 'SERIES'.

To find the number of three letter words that can be formed from the word 'SERIES', with or without meaning and without repetition. The number of permutations if all letters were distinct = ...
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### Permutation of MATHSBOMBE :

Of all the possible word formed by the permutation ( including non-sensical word ). If you were to arrange all the words in alphabetical order, what would be the position of the word 'MATHSBOMBE ' ...
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### $S_X = \{f(x) = x : \text{ bijective} \}$. prove $S_x$ is isomorphic to $S_n$

Aright, to start $S_n$ is the Symmetric group and $S_X = \{x_1, x_2, \ldots x_n\}$. Going through the mapping $\phi(S_X) \to S_n$, I'm not sure how I'd show this mapping and the first thought that ...
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### Alternating group on infinite sets

It is well known that the only normal subgroup of $S_n$ is $A_n$ when $n\geqslant 5$, and that $A_n$ is also simple. Furthermore, $A_{\infty}$, the even permutations on $\mathbb{N}$, is also simple. ...
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### Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
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### oligomorphic subgroups of $S_\infty$

Is it true that every oligomorphic subgroup of $S_\infty$ is not abelian? A subgroup of $S_\infty$ is said oligomorphic if its action on $\mathbb N^n$ has only finitely many orbits for each ...
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### Permutations and Combinations: Word Problem

You are to create a committee of $2$ men and $2$ women from a group of $8$ men and $8$ women. Each committee member will fill a role; president, vice president, secretary, and treasurer. How many ...