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

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1answer
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Counting techniques

I am preparing for an exam and I came across this problem. I am a little confused. Give the expression of ways to distribute 15 distinguishable balls into five distinguishable boxes so that the boxes ...
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3answers
137 views

combinatorics implementation in real life problems

How many ways there are to organize $7$ men in a row, if two insist on not standing next to each other? How do I approach this?
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1answer
79 views

Find a permutation

For $x = (12)(34)$ and $y = (56)(13),$ find a permutation $a$ such that $a^{-1}xa = y$. I written $a^{-1}xa = y$ as $xa = ay$ thus $(12)(34)a = a(56)(13)$ but I can't find the $a$?
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2answers
202 views

France Olympiad Team Selection Test 2005

In an international meeting of n ≥ 3 participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken by more than half of the participants. ...
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1answer
40 views

Product of non-disjoint k-cycles

I have two $k$-cycles $\alpha=(a \dots c \dots b \dots)$ and $\beta=(a \dots b \dots c \dots)$ and $\alpha \neq \beta^{-1}$. How to show that the product $\alpha \beta$ does not result in a cyclic ...
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4answers
983 views

In how many ways can we choose 5 balls from the box so that we have at least one blue ball?

A box has ten blue balls numbered from 1 to 10 and ten green balls numbered from 1 to 10. In how many ways can we choose 5 balls from the box so that we have at least one blue ball? I'm trying to ...
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1answer
227 views

Circular Nonconsecutive Permutations

A carousel has eight seats, each representing a different animal. Eight girls are seated on the carousel facing forward (each girl looks at another girl's back). In how many ways can the girls change ...
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1answer
32 views

Unkown number of objects.

If the number of permutations of $n$ different things taken $4$ at a time in which one particular thing does not occur is equal to that in which it does occur, find $n$. My teacher says $n=8$. I ...
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3answers
71 views

Permutation query involving 3 people with a limit of 45 as the sum.

I'm looking to find all the combinations of numbers possible in the below example. I shall do my very best to explain the situation clearly as I'm not too sure how to derive this outcome myself. ...
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1answer
87 views

Find permutation index of multiple lists where corresponding list indices match

I have several date time values: Mon 17h10 Tue 20h30 Wed 21h45 that maps to the following lists ...
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2answers
110 views

Combinations of the word REMEMBRANCE

4 letters must be chosen from the word REMEMBRANCE How many different selections can be done if there is no M, no R and at least 2 Es? I've seen this question somewhere but i don't remember the ...
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3answers
85 views

arrangement of a group of people.

The no. of ways in which 4 particular persons A,B,C,D and 6 other persons can stand in a queue so that A always stands before B, B before C and C before D is ? My try: since A B C D always have to ...
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2answers
132 views

permutations rolling die 6 times [closed]

How would you calculate the probability that if you roll a six sided die six times you will roll 1,2,3,4,5,6 consecutively. I am totally lost on how to even calculate this.
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1answer
126 views

Finding n in equivalence between two permutations

Find the value(s) of n: 2P(n, 2)+50 = P(2n, 2) For 2P(n, 2)+50 I simplified to 2(n)(n-1)+50. For P(2n, 2) I can't get any simpler than $2n\times(2n-1)\times...\times(n+1)\times n\times(n-1)$ and I'm ...
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1answer
404 views

Calculating the expected profit with Probability A level maths CIE

Company sets up display of 20 fireworks! for each firework, the probability that it fails is 0.05,independently of other fireworks the probability that more than 1 firework fails is 0.264 the 20 ...
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1answer
95 views

How many ways is there to do a round trip if at least one of the roads taken on the return trip is different?

I'm stuck on part c) of this question. The answer key gives 182. I already know there are 14 ways to make the trip from city A and city B and vice versa. It appears 182 came from $14 \times 13$ but I ...
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1answer
253 views

Why are we Multiplying here instead of Adding?

Three small towns designated by $A$, $B$ and $C$ are interconnected by by a system of two-way roads as described below [hopefully the following answer is enough info without picture] part a)In ...
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1answer
3k views

The product (or composition) of permutation groups in two-line notation?

This question has been asked before, I know: Product of Permutations However, his did not resolve my problem. Here's an example I've been looking at, which is to find the product of two ...
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1answer
62 views

Inner automorphisms of $S_3$

How do I prove that $S_3 \simeq \wp(S_3)$? So I must show that the group of inner automorphisms of $S_3$ is isomorphic to $S_3$. I haven't been given many examples on how to do these types of ...
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0answers
72 views

number of ways to arrange

There are N 1s and N 0s We have to arrange them in a row such that at no position in this row the number of 0s from the beginning exceed the number of 1s from the beginning. Also the number of ...
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1answer
108 views

What is this the name of this idea? (combinatorics)

The problem: There are three screws, each one a different type {Phillips, Robinson, Slotted}. There are three sets of screwdrivers, each set corresponds to a type of screw. There are no two ...
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0answers
243 views

Every group is isomorphic to a group of permutations?

Theorem: Every group is isomorphic to a group of permutations. Proof: Would Cayley's Proof from 1854 suffice the proof to this theorem?
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1answer
205 views

Permutations and combinations ! 9 different fruit pies divided between three

different fruit pies are divided between 3 people so that each person gets and odd number of pies .FInd the number of ways this can be done?? hint- so many combinations are added to get this answer .. ...
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2answers
382 views

Number of shortest routes

Suppose you have a wire mesh which is N by M units long, made up of unit square with wire at the edges. (So there are N+1 parallel wires all M long and, perpendicular to these, M+1 all N long). An ...
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1answer
68 views

How many ways to arrange $20$ items on $4$ towers

Suppose you have $20$ different rings and $4$ display towers. On each tower the rings are stacked one above another. In how many ways can they be arranged if: [a]: The order of rings on each tower ...
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1answer
101 views

How do I do cycle notation?

Consider the three permutations given in cycle form: $f = (124)(35)$, $g = (13)(45)(2)$, $h = (1)(4253)$ Which composition of the functions $(f, g ,h)$ gives the permutation $(1)(2)(34)(5)$?
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3answers
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In how many ways can four students be chosen from a group of 12 students?

Myself and my Math teacher are at a disagreement in to what the proper method of solving the question In how many ways can four students be chosen from a group of 12 students? is. The question comes ...
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0answers
83 views

number of ways to obtain a given permutation from k swaps

Let $\sigma_1, \ldots, \sigma_b \in S_n$ be all the 2-cycles ("swaps") in $S_n$. (So, $b = \binom{n}{2}$.) Given some $\pi \in S_n$, is there a known formula for how many ways to obtain $\pi$ as a ...
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2answers
272 views

$A_5$ has no subgroup of order 15 and 20

Show that $A_5$ has no subgroup of order 15 and 20. I have been thinking about this problem for so much time but I'm still clueless. Can anyone tell me how to do this problem? Thanks. I ...
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2answers
306 views

No subgroup of $A_4$ of order $6$, hence $Z(A_4)$ is trivial

I have a problem that I don't know how to solve. They are all specifically about Lagrange's Theorem and permutation groups. Can anyone help me? Thanks. Show that the fact that $A_4$ doesn't have ...
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4answers
193 views

Understanding the difference between combinations and permutations

Question: There are 6 men and seven woman in a club. A committee is to be formed. How many ways can we select a committee of five persons? Answer: C(13,5) or 13! / ((8!)(3!)) That is the answer ...
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2answers
160 views

Combinations and Permutations. Number of integer solutions

Question: How many integer solutions of $x_1 + x_2 + x_3 + x_4 = 17$ Satisfy: $x_1 \ge 0 , x_2 \ge 1, x_3 \ge 2, x_4 \ge ...
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1answer
125 views

Properties of Permutations of a Set A

Let $A$ be a finite set, and $B$ a subset of $A$. Let $G$ be the subset of $S_A$ consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove that $G$ is a subgroup of ...
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1answer
332 views
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Assumption in proof: The alternating group $A_n$ is simple.

Let $N$ be a nontrivial normal subgroup of $A_n$, $n \ge 5$, where $A_n$ denote the alternating group. The book want to prove that $N$ contains a 3-cycle. (Niels Lauritzen, Concrete Abstract ...
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3answers
213 views

Proving that two permutation groups are isomorphic

Here's the statement to prove: Let $n,m$ be two positive integers with $m≤n$. Prove that $S_m$ is isomorphic to a subgroup of $S_n$, where $S_n$ is the collection of all permutations of the set ...
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2answers
37 views

Subgroup $\{(1),(12)\}$ in $S_3$ is not kernel of any homomorphism

How do I prove the following statement Subgroup $\{(1),(12)\}$ in $S_3$ is not kernel of any homomorphism.
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3answers
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Counting more strings with 7 letter

Already made one sort of like this earlier (Counting strings with 7 letters), but I'm still not getting into the mindset required for this kind of tasks. Anyway, I'm given the letters A-G and.. I ...
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1answer
41 views

Unlimited Exercise

I have 5 groups with 3 different exercises (i.e. group 1 has pushups, pull-ups and dips). I am to choose one exercise from each group to make a "round". So how many rounds can I come up with using ...
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2answers
404 views

Find number of solutions of the equation x1+x2+x3 = 41, where x1, x2 and x3 are odd and non negative integers

There are two constraints to this problem: 1) x1, x2 and x3 are non negative integers 2) x1, x2 and x3 are odd If there had been just the first constraint (non negative integer), i would have ...
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4answers
432 views

In how many ways can $7^{13}$ be represented as product of $3$ natural numbers?

How i solved it: all possible non-distinct groups $(a,b,c)$ are, $a = 0 \Rightarrow (b,c) = (0,13)(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)$ $a = 1 \Rightarrow (b,c) = (1,11)(2,10)(3,9)(4,8)(5,7)(6,6)$ $a ...
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3answers
71 views

Problem with permutations

The problem says: We have strings formed by two letters, followed by two digits and then followed by three letters. In each group repetitions are not allowed, but the last group of three letters ...
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1answer
64 views

2 regular graphs and permutations

I have found this question on MSE before but I didn't find the answer satisfactory and it is so old I doubt anyone is still following it. Let $f_{n}$ be the number of permutations on $[n]$ with no ...
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1answer
251 views

Baseball Roster Optimization

I'm trying to programmatically optimize a Fantasy Baseball Roster that requires a fixed number of players at position (2 Catchers, 5 Outfielders, etc.) and has a salary constraint (total draft price ...
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1answer
116 views

Does a homomorphic image of even permutations consist of even permutations?

If $f:S_n \to S_n$ is a homomorphism, prove $f(A_n) \subseteq A_n$. If every image of a transposition is even, then there is nothing to prove, but it is not sure.. How can I prove the problem?
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120 views

Permutation: How many ways to put 7 people in 10 rooms?

How many ways can 7 people be placed into 10 rooms, if (only) 2 of them can’t share a room with anyone? I'm not sure how to go about this, mostly because of the "share a room" bit. I'm thinking I ...
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1answer
41 views

Group Theory of order in $S_{10}$

What is the order of $(3\ 7)(4\ 5)$ in $S_{10}$? Answer: Order $2$ Is my answer correct? Thank you in advance
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1answer
94 views

Order of a product of two cycles

Is it true or false? If $a$ is a permutation that is an $m$-cycle and $b$ is the permutation that is $n$-cylce then order of $ab = \operatorname{lcm}(m,n)$.
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Index of centralizer of alternating group

For any element $x \in A_5$, we have that $$[A_5:C_{A_5}(x)]=\begin{cases} [S_5:C_{S_5}(x)], & \text{condition 1} \\ \frac{1}{2}[S_5:C_{S_5}(x)], & \text{condition 2} \end{cases}$$ Basically, ...
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1answer
102 views

A few questions relating to counting for midterm practise exam?

I'm doing some questions for my midterm practise exam (multiple choice) for discrete structures and would appreciate some help (My answer is bolded): Using the 26-letter alphabet {a,b,c,...,z}, how ...