# Tagged Questions

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

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### $S_6$ contains two subgroups that are isomorphic to $S_5$ but are not conjugate to each other

This is a problem from Ph.D. Qualifying Exams. Show that the symmetric group $S_6$ contains two subgroups that are isomorphic to $S_5$ but are not conjugate to each other. Here is my method. $S_5$ ...
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### Conjugation of permutation group $S_n$

I want to find the conjugacy classes of the permutation group $S_n$ To start with I think I have to prove that $\pi(\sigma_1\dots \sigma_m)\pi^{-1} = (\pi(\sigma_1)\dots \pi(\sigma_m))$. Where $\pi$ ...
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### How to find a permutation $\sigma$ given the permutation $\sigma^2$?

How to solve the equation: $\sigma ^2 =\left({\begin{array}{*{20}c}1 & 2 & 3 & 4 & 5\\ 1 & 4 & 2 & 3 & 5\end{array}}\right)\$ where $\sigma \in S_5$. Is there a ...
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### number of combinations/permutations

if I have $n$ drawers and in each drawer I can only have 1 pen or 1 pencil for example if i have $3$ drawers the possible ...
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### Probability of picking up one ball of each color

A box contains 6 red, 4 white and 5 black balls. A person draws 4 balls from the box at random. Let P be the probability that among the balls drawn there is at least one ball of each color. Find 455 * ...
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### Possible number of sequences such that $x_{i} = 1$ or $2$ and $\sum_1^n x_{i} = 10$

How many finite sequences are there such that $x_{i} = 1$ or $2$ and $\sum_1^n x_{i} = 10$ $?$ Now I did it this way: number of $1$'s $\$:$\$ $0$ ,$2$ ,$4$ ,$6$ , $8$ , $10$ ...
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### Permutations of {1, 2 .. 30} where $a_n - a_n-m$ is divisible by m from {2, 3, 5}

There are $N$ permutations $(a_1,a_2,\dots,a_{30})$ of $1,2,\dots,30$ such that for $m\in\{2,3,5\}$, $m$ divides $a_{n+m}-a_n$ for all integers $n$ with $1\leq n <n+m\leq 30$. Find $N$. I really ...
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### How many structurally different latin squares of order 5 do exist?

I know the number of latin squares order 5 which start with 1 2 3 4 5 in the 1st row or column, that is 1344, but the greater part of that number consists of structural duplicates of each other. So, I ...
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### If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
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### Permutation count of AABBC

Given a string: $AABBC=A^2B^2C^1$ I am trying to find the Total Permutations (this may be incorrect): $\dfrac{5!}{2!\cdot2!}=30$ My question is how would I find the partial sums (perhaps the ...
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### How many two letter words can be formed from 26 English letters?

There are 26 English letters(a-z). From layman approach, How can one calculate the possible two letter words from these 26 English letters?
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### Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
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### How to conceptualize “dividing out” a number (e.g. in permutations, Bayes' Theorem)?

I'm trying to achieve a better conception of what it means to "divide out" a variable/number, because I'm currently have a lot of trouble justifying to myself why it actually works the way it does in ...
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### Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
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### How many permutations of the word TOMORROW can be made if the O's can't be together?

I'm trying to answer this question. This is my attempt of solution: First we distiguish the O's and R's, then we have the word: $TO_1MO_2R_1R_2O_3W$. We have $8!-7!\cdot3!-6!\cdot 3!$ different ...
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### Combinatorics Question - Permutations and Supersets

I had a question that seems pretty straightforward, but I can't seem to wrap my mind around it. Let's say I have a bunch of elements in a set. {A, B, C, D, E}. How many permutations are there of ...
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### Permutation of numbers from multiple sets [May contain duplicate numbers among other sets], resulting in Non-Duplicate Set

We have 3 Data Sets. From each set we will be selecting few numbers. 3 from Set 1, 2 from Set 2, 3 from Set 3. Totally, we will get 8 Numbers from 3 Sets. The resulting sets shouldn't contain any ...
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### Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
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### Characterisation of the squares of the symmetric group

I found out that for $n\le 4$ we have $S_n^2=A_n$ with $G^2$ defined by $$G^2:=\{g^2 \mid g\in G\}$$ for any group $G$. Surely we have $S_n^2\subseteq A_n$ for all $n\in\mathbb N$. Is there a ...
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### Show that there is always a way to achieve det(A) > 0

a) Assume that $(a_1, ..., a_9)$ are different positive numbers. Let us make a $3 \times 3$ matrix $A_s$ by placing them arbitrarily into $9$ positions available. Show that there is always a way to ...
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### Proving a certain lemma about subgroups of $A_n$

In proving $A_n$ is simple for $n\neq4$, my teacher established the cases 1, 2, 3 as obvious, then proved the case 5, and proceded by induction on the rest. In the midst of that induction, he stated ...
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### How to arrange 15 women and 15 men so no two females are seated next to each other?

To a certain conference, each firm can send two employee representatives, on the condition that one of them is a male and the other a female. If 15 firms were represented in this conference, what is ...
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### When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...