# Tagged Questions

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

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### Better way to compute commutator subgroup of $A_n$ for $n\geq 5$

I want to show that $[A_n,A_n]=A_n$ for $n\geq 5$. Clearly $[A_n,A_]\leq A_n$. But to show the reverse inclusion I have an answer which involves too much "calculation"(As $A_n$ is generated by the ...
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### Recurrence forlmula for number of permutation.

The problem: Find the recurrence formula for number of permutations if a cube of any such permutation is identity permutation. Solving: We have to count a number of permutations kind of $\pi$, if ...
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### If $G$ acts such that each non-trivial element either has no fixed point or exactly two, then there exists fixed-point free involutory map on $\Omega$

If a finite group $G$ acts on a set $\Omega$ non-regulary (i.e. there is some element fixing some point) and each element having some fixed point has exactly $n$ fixed points, then we say the group is ...
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### permutations of binary sequences

What is the proof that there are $2^n$ distinct binary codes of length n I know this progression also applies to the decimal ($10^n$) and hex ($16^n$) systems but how can this be shown?
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### Theorem about non-regular group action, where an element fixes no points or exactly $p$ points

I have a question on the following proof. All groups are assumed to be finite. But first I will mention a lemmata: Lemma: Let $G$ act faithfully and non-regular as a group such that there exists some ...
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### ii.) Probablity of $2$ bags containing white and black balls.

A bag contains $4$ white balls and $2$ black balls , another contains $3$ white balls and $5$ black balls . If one ball is drawn from each bag, determine the probability that one is white ...
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### How many permutations of $n$ elements are there which move $k$ elements?

How many permutations of the elements $[x_1,\ldots,x_n]$ are there which actually move $k\leq n$ elements and keep $n-k$ elements at the same position? What I mean by move is that, if element $x_i$ ...
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### Probability of $2$ bags containing white and black balls.

A bag contains $4$ white balls and $2$ black balls , another contains $3$ white balls and $5$ black balls . If one ball is drawn from each bag, determine the probability that both are ...
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### Probability of having at least one pair by drawing 4 shoes from 12 pairs.

There are $12$ pairs of shoes in a cupboard. $4$ are drawn at random. What is the probability that there is at least one pair? My first attempt: If we chose a pair at first and then draw any two at ...
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### The probability that k fixed elements of the set belong to one cycles in random permutation

Given the set of n elements: $\mathbb{A} = \{1,...,n\}$. Fix $k$ elements in the set. And consider a random permutation $\pi : \mathbb{A} \rightarrow \mathbb{A}$. So any permutation can be ...
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### How many ways can 12 different pennies be distributed to four people without each person getting exactly 3?

Question: How many ways can $\mathbf{12}$ different pennies be distributed to four people without each person getting exactly $\mathbf{3}$? My thoughts: I'm really not too sure how to approach this ...
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### binary combinations/rule of sums

I'm currently working on this topic but I'm having a hard time. In an 8-bit string, there are 256 combinations (or is it permutations?). Either way, I know it is 2^8. If at least 3 elements must be ...
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### Find the prob. to win third prize in lottery game

I have a question about prob. The question is "A lottery game that chooses 5 numbers out of 50 numbers. What is probability to win the third prize which is the case matching 3 out of the 5 numbers " ...
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### Group Theory, conjugation of permutation group

I've been given the following question in the context of group actions through conjugation but I'm having difficulty understanding what is being asked Let $\tau$ be any permutation in $S_m$. Let ...
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### Determine the total number of $4$-digit numbers which can be obtained using the digits $1, 2, 3, 4, 5$.

It's a basic question, but I don't know why I am getting confused. Determine the total number of $4$-digit numbers which can be obtained using the digits $1, 2, 3, 4, 5$. Also find how many of them ...
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### Cycle index of a symmetric group defined by recurrence relation

It is claimed in a text I am reading that the cycle index of the symmetric group satisfies the recurrence relation $$Z(S_n)=n^{-1} \sum_{k=1}^n{s_kZ(S_{n-k})}$$ This is presented without explanation, ...
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### Let $A \unlhd G$ and $o(x) = 3$ for each $x \notin A$, then $[B, B^x] = 1$ for abelian subgoups $B \le A$.

Let $A$ be a normal subgroup of $G$. If each element from $G \setminus A$ has order $3$, then $[B, B^x] = 1$ for every abelian subgroup $B \le A$ and $x \in G \setminus A$. Any hints for this ...
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### If 36 boxes can only hold one ball, how many ways are there to arrange 21 red balls and 15 blue balls in them?

Given 36 boxes that can only hold one ball each, how many ways are there to arrange 21 red balls and 15 blue balls in them? (Pretty new at this).
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### Finding the next k-permutation of n

I have a set of numbers say $\{x_1, x_2, x_3, x_4, x_5, ... x_n\}$ and a number $k$. We can form $n!/(n-k)!$ permutations using this. My question is: Suppose $n = 5$ and the set of numbers is like ...
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### Probability - distribute people in a bus

(a) In how many ways can 6 people be lined up to get on a bus? Answer = 6! (b) If 3 speciﬁc persons, among 6, insist on following each other, how many ways are possible? Answer = 4! * 3! (c) ...
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### How many ways can 10 identical buttons, 10 identical bows, and 10 identical beads be distributed to 4 different people?

Question: How many ways can 10 identical buttons, 10 identical bows, and 10 identical beads be distributed to 4 different people? My thoughts: I decided to use the bars and stars method. With the ...
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### Total number of combinations for pairing from two different sets

Assume we have two sets of numbers {1, 2, 3} and {4,5,6}. My aim is to pair two numbers together from the two different sets such as {1} with {4} {2} with {5} {3} with {6} another pairing ...
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### How many combinations of passwords can be created according to these conditions

Suppose that a password must have at least $8$, but not more than $10$ characters, where each character in the password is either one of the $26$ lower case English letters, or one of the $10$ ...
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### How many ways can 4 men and 4 women stand in a line so that the men are together and the women are together

Question: How many ways can $\mathbf{4}$ men and $\mathbf{4}$ women line up with all the women together and all the men together? My thoughts: I begin my solution to the problem by adding the total ...
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### Pizza Combinatorics Question

How many possible combinations of pizza,dips are there when? $4$ pizzas with up to $6$ toppings each, $10$ to choose from $4$ kinds of dip, $7$ to choose from Each pizza cannot have the same ...
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### Does every automorphism of a permutation group preserve cycle structure?

In class we proved that the conjugacy map preserves cycle structure, and I was wondering if this was the case for any automorphism of a permutation group. I intuitively think that it should be the ...
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### (Enumeration) In how many ways can 5 people be placed in 8 rooms

In how many ways can 5 people be put into 8 rooms if only 2 of the people (very dubious characters) can’t share a room with anyone? Note that there is no maximum capacity for the rooms. My attempt: 2 ...
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### A norm invariant under permutations but not under signed permutations

Let $\| \cdot \|$ be a norm on $\mathbb{R}^n$. We call it axes-symmetric if $\|x\|$ does not depend on the order of the components of $x$. Equivalently if $\|x\| = \|P \cdot x \|$ for any permutation ...
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### permutation with repetition subset

How many distinct strings of length 4 can be generated with $c,b,b,a,a,d$ Through a script I know that there are 102 such possibilities. My Attempt Case 1: using only one 'b' and one 'a'. This can ...
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### Permutations in a closed loop

Here's an example of what I mean by permutations in a closed loop: A necklace is to be made by threading four identical black beads and four identical white beads onto a string which is closed ...
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### Let $G$ be a solvable primitive permutation group. Why the degree of $G$ is a prime power

Let $G$ be a solvable primitive permutation group. Why the degree of $G$ is a prime power and $G$ has a unique minimal normal subgroup? (8B.4 problem of Finite group theory by Issac) Is ...
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### Prove that the set $U_1 =\{(123),(124),…,(12n)\}$ generates $A_n$

Here $A_n$ = the alternating group on n symbols. We are given a hint: $(abc) = (1ca)(1ab)$ $(1ab)=(1b2)(12a)(12b)$ $(1b2)=(12b)^2$ I am unsure where to start. Can someone give me a jump start?
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### How can I show that the following pairs of permutations are in the same conjugacy class in S5:

How can I show that the following pairs of permutations are in the same conjugacy class in S5? (1,2,3,4,5) and (1,5,2,4,3) (1)(5,3)(2,4) and (2)(1,3)(5,4) (1,3,2) and (1,4,5)(2,3)
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### Algebraic proof that $\det AB = \det A \cdot \det B$ using Leibniz formula for determinants [duplicate]

The Leibniz formula for determinants allows us to express an $n \times n$ matrix determinant as a sum over permutations in $S_n$: \det A = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \cdot ...
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### Permutation: $5$ into $8$ if only $2$ can share

I am having trouble figuring out a permutation problem: "In how many ways can $5$ mathematicians be put into $8$ offices, where each mathematician has an office to themselves? What if only $2$ of ...
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### Probability: ATM Random Security Code Computer Generation.

In order to access one's bank account using an ATM, the user must key in a set of numbers making up a security code; a code can be anything from a 1-digit to a 5- digit number. This code is randomly ...