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

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Cycle structure of the generators of the dihedral group

Would it be correct if I say the following about the cycle structure of the generators of the dihedral group, $D_n$? If $n$ is even, the generator is $<(2 \hspace{0.5cm} n) (3 \hspace{0.5cm} n-1) ...
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44 views

Partition of natural number not equal to factorial

I wish to prove the following statement so I can use it as a lemma for a group theory result. To be honest I have not tried much yet, my intuition tells me this is going to be connected to the ...
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1answer
379 views

Determining Permutation/Combinations with Bit Strings

I've got a discrete math problem on my hands...I'm trying to understand the steps behind working with bit strings; specifically, how a bit string of x length has "at least" or "exactly" a certain ...
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1answer
18 views

permutations of n objects

Does the number of permutations of $n$ objects, $r$ alike of one kind and $n−r$ alike of another kind, always equal the combinations of n different objects taken $r$ at a time? Explain. I know ...
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15 views

Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given ...
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1answer
80 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
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1answer
53 views

Simple Question on Binomial theorems… [on hold]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
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4answers
7k views

how many 5 digit numbers are there with distinct digits?

I found this question on gre forum, it's answer was given by this expression: $9\cdot9\cdot8\cdot7\cdot6$ which I heard in school as well. What I tried to do was: for numbers from index $4$ to ...
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529 views

Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...
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1answer
29 views

How many straight lines can be made between 10 points such that 4 of them are colinear?

So i know how to get the answer. We just have to find $C(10,2)$ and subtract $C(4,2)$ and add 1. We are basically counting all the points between co-linear points as 1. So the question is why we are ...
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3answers
64 views

How many distinct ways can the number be written as product of $3$ factors?

How many distinct ways can the number $126$ be written as a product of $3$ positive integer factors? I found that the prime factors are $126=2\times3\times3\times7$. But how to get number of ...
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1answer
19 views

For each of the following restrictions, find the smallest size n for strings over $\{a, b, c\}$ that can be used as codes for $27$ people.

For each of the following restrictions, find the smallest size $n$ for strings over $\{a, b, c\}$ that can be used as codes for $27$ people. a. There are $k$ $a$’s, $l$ $b$’s, and $m$ $c$’s and $k + ...
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2answers
37 views

What is probability that out of the first half on N objects, none will be matched with their own label?

The problem: We have N (even) objects ordered $o_1 ... o_N$ , each having their own label. The labels are reassigned to the objects randomly. What is the probability that that neither of the first ...
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2answers
94 views

If we are given that a list of $n$ numbers has $11,660$ derangements, what is the value of $n$?

The Full Question For the positive integers $1,2,3,\dots n-1,n$, there are $11,660$ where $1,2,3,4,5$ appear in the first five positions. What is the value of $n$? My Work First I considered all ...
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1answer
21 views

Understanding derangement.

From the inclusion-exclusion principle we get that out of $N$ objects with one label each, there is a probability of $$\sum_{k=1}^N (-1)^{k+1}\frac{1}{k!}$$ that a random assignment of the $N$ labels ...
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23 views

Centralizer of $\sigma\in S_n$ [duplicate]

Let $\sigma\in S_n$ . Describe the centralizer of $\sigma$. Thought: If I conjugate $\sigma$, then $\tau^{-1}\sigma\tau=\sigma.$ This means that $\tau$ is a power of $\sigma,$ and ...
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1answer
24 views

How many ways are there to arrange the letters of word $ALGEBRA$ such that the relative order of the vowels and consonants doesn't change?

I did this question this way :- there are 4 consonants in the words (LGBR) and there are 7 letters in the word. $therefore$ number of in which consonants can be arranged in relative order will be ...
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2answers
33 views

A question of permutations and combinations with six cards and six envelopes.

Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same ...
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4answers
34 views

How many mixed double pairs can be made from 7 married couples provided that no husband and wife plays in a same set?

So for first man there can be 7 possible partners including his wife, for the next man there will be 6 possible partners and so on, $therefore$ for $7$ men and $7$ women, there will be $7!$ possible ...
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3answers
272 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
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1answer
61 views

Number of permutations in lawn tennis so no husband and wife play together.

In how many ways can a lawn tennis mixed doubles be made up from seven married couples if no husband and wife play in the same set? Please explain the logic.
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3answers
76 views

Finding number of functions from a set to itself such that $f(f(x)) = x$

The questions states that $f: A\rightarrow A$ is a function which satisfies $f(f(x)) = x.$ We have to find the number of such functions with $A = \left\{1,2,3,4\right\}$. The given condition clearly ...
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2answers
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Counting the number of “distinct” permutations of two sets?

I don't really know how to introduce this question, so I start defining something I needed in order to well understand the problem I met! Let $A$, $B$ two finite sets of distinct elements, with ...
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63 views

math software - permutation group elements operation

I need a software allows to calculate operation elements of permutation group. For example the following elements operation yields identity permutation $$ (1234)(1423) = (1)$$ Sage seems to solve the ...
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How do I calculate the number of unique permutations in a list with repeated elements? [duplicate]

I know that I can get the number of permutations of items in a list without repetition using (n!) How would I calculate the number of unique permutations when a ...
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1answer
30 views

rationale for book's solution of combinatorics question about scheduling ten speakers with restrictions

If A, B, C are among $10$ people speaking at a function in alphabetical order What are total ways of doing so. BOOKS APPROACH: There are $10$ people out of which $3$ need to be taken care of. So ...
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43 views

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A?

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A? I have a doubt in the selection of A at the last position. Please help. Thanks in advance!!
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1answer
69 views

Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
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1answer
31 views

Permutations of 3 digit numbers divisible by 5

I recently had to answer the following permutation question: How many 3 digit numbers can be formed from the digits 2,3,5,6,7,9 which are divisible by 5 and none of the digits are repeated? ...
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66 views

Setting up an inclusion-exclusion question

What is the number of one-to-one functions $f$ from the set $\{1,2,...,n\}$ to the set $\{1,2,...,2n\}$ so that $f(x) \neq x$ and $f(x) \neq 2n - x + 1$ for all $x$? I'm getting that the number of ...
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In how many ways can the letters of word $PERMUTATIONS$ be arranged if there are always 4 letters between P and S?

In how many ways can the letters of word $PERMUTATIONS$ be arranged if there are always $4$ letters between $P$ and $S$? Now there are $12$ blank spaces, which we have to fill by the letters of ...
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1answer
35 views

Combination and Permutation S= {A,B,B,C,C,C,D,D,D,D,E,E,E,E,E}.

S= {A,B,B,C,C,C,D,D,D,D,E,E,E,E,E}. If I choose n element from S, how many possible combination (unordered) and permutation (ordered) are possible (without using decision tree or counting)? What is ...
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394 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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32 views

Combinatorics Puzzle (Circular Table) [on hold]

Q. Eleven members of a cricket team are numbered 1,2,3..11. In how many ways can they be seated around a circular table so that the numbers of any two adjacent players differ by one or two. ...
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1answer
45 views

Arrangement of letters in VISITING with no pairs of consecutive Is.

Could someone help me understand a book's solution to the following problem? I am providing my own solution, but I fail to understand theirs. Question: How many ways are there to arrange the ...
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1answer
440 views

binary strings and counting sequence problem

Hello i'm working on these questions and I have few questions 1) A binary string is a finite sequence of 0 and 1. Ex. 001101 is a string of length 6 a) List all binary strings of length 4 (so I ...
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Number of letters moved by a product of permutations

Let p and q be permutations in the symmetric group on n letters. p and q need not have the same cycle structure. Now compute q * inv(p) -- for inv(p) the inverse of p -- and count the number of ...
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1answer
13 views

Permutations starting with a specific letter

Ok, this is a homework question and I think I've resolved it but I want to bounce it off you guys. I have a $6$ letter word with no repeated letters. I need to calculate how many $3$ letter words can ...
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1answer
60k views

How many possible combinations in 8 character password?

I need to calculate the possible combinations for 8 characters password. The password must contain at least one of the following: (lower case letters, upper case letters, digits, punctuations, special ...
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2answers
29 views

$5$ chem students, $6$ maths students and $7$ physics students permutation

$5$ chem students, $6$ maths students and $7$ physics students. Find the number of arrangements if a)Chem majors are to occupy the first 5 positions b)Chem majors cannot occupy the first 5 positions ...
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1answer
42 views

Painting a 2x2 Grid

We have a 2x2 grid and 10 different colours. I want to paint such that adjacent grids are painted with different colors. How many ways can i do this? ...
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38 views

Positive integers $<100000$, how many contain exactly one $3$, one $4$ and one $5$

So I use $5$ positions for range $00000$ to $99999$ Choose $3$, choose $4$ and choose $5$ as follows: $5C1 \cdot 4C1 \cdot 3C1$ Remaining $2$ digits have $7$ possible digits as input Ans: $5C1 ...
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25 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ ...
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63 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
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2answers
28 views

If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
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47 views

Prove that the funtion f: $G\rightarrow G$, defined by $f(x)=x^k$, $x \in G$ is a permutation of $G$

Help me with this exercise, I could not do it :( Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the function f: $G \rightarrow G$, defined by ...
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30 views

Write $π = (3, 2, 5)(2, 5, 4)$ in “table” notation?

Isn't this impossible...? Because this permutation goes from 3 --> 2 ---> 5 ---> 3 according to the first cycle, but goes from 2 --> 5 ---> 4 ---> 2 according to the second cycle. So 5 can't go to 3 ...
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18 views

Permutation of the alphabets of the word “mediterranean” such that first and fourth letter are “r” and “e” respectively.

Above is the original question. The correct answer is in green that is 59. I have chosen option 3 that is $\frac{11!}{(2!)^3}$ because I thought that there are 13 alphabets in the word ...