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

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Permutation & Combinations - Distribution

The number of ways in which n distinct things can be distributed among n people so that at least one person does not get anything is 232. Find n. I think every object has (n-1) option. So (n-1)^n=232. ...
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2answers
67 views

Find the number of ways this can be arranged in which no 2 women and no 2 men sit together given 4 men and 3 women are seated in a dinner table?

Find the number of ways this can be arranged in which no 2 women and no 2 men sit together given 4 men and 3 women are seated in a dinner table? @Edit: They are seating in a row dinner table I have ...
3
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1answer
59 views

Group $G$ acting on $\Omega$ such that each $\alpha \in \Omega$ has unique $p$-element fixing $\alpha$.

Let $G$ be a group acting on a set $\Omega$ and let $p$ be a prime. Suppose that for each $\alpha \in\Omega$ there is a $p$-element $x \in G$ such that $\alpha$ is the only point fixed by $x$. If $\...
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3answers
524 views

In how many words the letter of word RAINBOW be arranged so that only 2 vowels always remain together?

My Approach: RAINBOW has 4 Consonants and 3 vowels. Out of 3Vowels 2 vowel are selected and arranged in 3P2 ways and the rest letters are arranged in 5! ways(1vowel and 2 consonants) The ...
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2answers
271 views

In how many ways the letters of the word RAINBOW be arranged such that A is always before I and I is always before O.

I research some sites and books and i found these this approach helpful but could not understand a bit. Approach: All the 7 letters of the word can be arranged in 7! ways. and 3particular letters ...
3
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3answers
154 views

Number of Non - Decreasing functions?

Let A={1,2,3.....10} & B={1,2,3....20}. We have to find the number of non decreasing functions from A-->B. What I tried :No. Of non decreasing functions = (Total functions) - (Number of ...
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1answer
954 views

Find the sum of all 4 digit numbers which are formed by the digits 1,2,5,6?

I have researched and found 2 approaches but haven't understood both.Can anyone explain it clearly or probably with any real world example? Approach 1 Four digit numbers $ = 4 \cdot 3 \cdot 2 \...
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1answer
41 views

Permutations where no partial sum is divisible by 3 (contest question)

A permutation of the integers $1901,1902\dots 2000$ is a sequence in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums $$s_1 = a_1,\...
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2answers
69 views

Correctly calculating permutations and combinations without duplicate patterns

Given 16 balls each numbered 1 through 16, and 5 glass tubes numbered 1 through 5; how many ways are there to slot all 16 balls into the glass tubes, selected one at a time, with the only condition ...
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3answers
75 views

In $S_7$, find the permutation $x$ that satisfies the condition $x^2 = (1,2)(3,4)$.

In $S_7$, $X^2 = (1,2)(3,4)$. What is the permutation $X$ that satisfies the given condition?
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1answer
73 views

How to understand the set of permutation representations of a group $G$?

In Algebra by Michael Artin, Chapter 6, page 182 (second edition, Pearson), Proposition 6.11.2 states that there exists a bijective correspondence between operations of a group $G$ on the indices set ...
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2answers
66 views

Conjugacy in $S_n$ with composing permutations left to right vs. right to left

I realize there are two conventions for composing permutations. Left to right: $(1\ 2)(1\ 3) = (1\ 2\ 3)$ Right to left: $(1\ 2)(1\ 3) = (1\ 3\ 2)$ Among others, Dummit and Foote and Contemporary ...
2
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0answers
108 views

A fashion victim puzzle

Consider $n \in \mathbb{N}$ fashion-sensitive kids, each wearing a T-shirt; for simplicity, kid $i \in \{1, \ldots, n\}$ initially wears shirt $i$. Tastes over the shirts are summarized in an $n \...
3
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1answer
60 views

Finding the n-th arrangement of items with repetitions [duplicate]

I'm new to Stackexchange and maybe I do not have the correct mathematical terms for the question I'm about to ask. I'm given a multiset of given size $N$ which consists of zeros and ones. Example: ...
2
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1answer
84 views

A Permutations/Combinations Question and Inquiry on Good Source for Studying The Concept

Lets say a burger joint offers options for customizing burgers. There are 3 types of meats and 7 condiments. A burger must include meat but may include as many or as few condiments as the customer ...
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0answers
123 views

How many anagrams of a given word exists with constraints

I saw many questions about anagrams here, but neither one fits my needs. Let's say we have the word MISSISSIPPI. I need to find the count for those anagrams that meet the criterias as follows: ...
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2answers
196 views

A three-character password, how many different passwords are possible?

a three-character password consists of 2 different digits between 0 and 9 inclusive. and 1 letter of the English alphabet. the letter must appear as first or second character, how many different ...
2
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1answer
29 views

Signum or parity of a transposition is $-1$

The definition of signum $\alpha$ is given by $$sgn(\alpha)=(-1)^{n-t}$$ where $\alpha=\beta_1\dots\beta_t$ a complete factorization of disjoint cycles. If $\alpha$ is a transposition, then it moves $...
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2answers
303 views

How many permutations of the letters AEIOU contain the strings EA and UO?

So here is our "word" AEIOU. Then we need to find how many permutations contain EA and UO. Then how many contain AE and EI and how many end with O. I know how to figure out some of these problems ...
0
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1answer
62 views

Basic probability - lottery type question

I know this might sound really basic, but my maths/stats knowledge has gone very rusty over the years... Maybe I can get a head start here. If I ask my user to choose 5 numbers, how do I calculate ...
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1answer
29 views

Total number of possible permutations [closed]

How many ways can 10 people be seated in a row, so that a certain 2 of them are not seated next to each other.
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69 views

Composing a permutation with a transposition and length

Let $\pi$ be an element of the permutation group $S_n$, such that, when it operates on the ordered set $\{1, 2, \ldots , N\}$, the ordered set that it produces $\{k_1, k_2, \ldots , k_N\}$ has two ...
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0answers
37 views

Minimal polynomial of endomorphism of permutation module

Let $G$ be a transitive permutation group on a set $\Omega$. If $n$ is the degree and $M\in\mathbb{Z}^{n\times n}$ is a symmetric matrix that is also contained in $\operatorname{End}_{\mathbb{Q}G}(\...
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589 views

Ordered Arrangement of Books

A shelf has $10$ distinct books of which $3$ are Math,$ 4$ are English and $3 $ are Science books. In how many ways can you arrange these books such that all English books will be placed in the center?...
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1answer
411 views

A team has 13 members suppose seven are women and six are men? How many groups?

I'm having trouble figuring out the permutations and how to properly multiply them together for this problem. A computer programming team has $13$ members. a.) How many ways can a group of seven ...
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0answers
135 views

Use of permutation/combination in Geometric problems

There are $p$ points in space, no four of which are in the same plane with exception of $q$, which are all in the same plane. Find out how many planes there are each containing three of the points? ...
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2answers
42 views

Clarification of a concept in Permutation

Statement 1 No. of ways in which $(m+n+p)!$ different things can be divivded into different groups containing m,n & p things respectively. is $(m+n+p)!/m!n!p!$ Statement 2 If $m=n=p$ and the ...
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4answers
147 views

How to find number of numbers formed with given digits?

Question is Find the number of numbers of five digits that can be made with the digits of the number 1203210. Can you please explain the problem? I did not understand it. Although one asked similar ...
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3answers
51 views

A problem on Permutation/Combination.

Q.How many ways no.s less than 10000 can be made with digits $1,2,3,0,4,5,6,7$? My attempt: The no.s should be of $4$ digit. It cant start with $0$ Then,the no.s of no.s possible should be $7*7*6*...
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2answers
145 views

Proof of a formula related to permutations and combinations involving geometrical figures

How to prove that the number of triangles that can be formed by joining the angular points of a polygon of n sides as vertices are$\dfrac{n(n-1)(n-2)}{6}$?
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103 views

Missionary and Cannibal problem

I got an interesting problem yesterday (Yes, for homework, but it seems like this is on topic) The problem goes like this: Three missionaries and three cannibals wish to cross a river. There is a boat ...
2
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1answer
64 views

There are 3 workers in a company that has 5 working days in a week.

There are 3 workers in a company that has 5 working days in a week.In how many ways can the 3 workers take leave/rest if no two workers can take leave on the same day. Attempt: The first worker can ...
0
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1answer
88 views

A problem in permutation/combination

Q.A boat is to be manned by eight men,of whom $2$ can only row on bow side and $1$ can only row on stroke side,in how many ways can the crew be arranged? I approached the problem in the following ...
0
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1answer
25 views

How to calculate route variations/permutations

If I have 5 trucks and 10 deliveries to make per truck, that's 50 deliveries total, but how many different route variations could there be? You could give each truck the list of deliveries and they ...
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2answers
100 views

Why does $\sum_{\sigma\in S_n}q^{\ell(\sigma)}=\frac{(1-q)(1-q^2)\cdots(1-q^n)}{(1-q)^n}$?

This is a known result, but I can't find a proof. Why does $$ \sum_{\sigma\in S_n}q^{\ell(\sigma)}=\frac{(1-q)(1-q^2)\cdots(1-q^n)}{(1-q)^n}? $$ Here $\ell(\sigma)$ is the length of $\sigma$, ...
0
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1answer
34 views

Sampling without replacement from unknown sample size

Five mice are chosen (without replacement) from a litter, three of which are tagged $A$, $B$ and $C$. The probability that all three tagged mice are chosen is twice the probability that $A$ is the ...
0
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1answer
30 views

Number of ways 1a,1b,5 can add up to n (with this being a permutation)

This problem is on my homework. A vending machine dispensing books of stamps accepts $\$ $ 1 coins, $ \$1 $ bills and $ \$5 $ bills. A) Find a recurrence relation for the number of ways to deposit n ...
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3answers
94 views

Pick $x$ out of $y$ objects. Match $n$ picks.

Computer randomly picks a group of 6 objects out of 30, no repetitions. User than picks $6$ objects out of those $30$, also no repetitions. What are the odds of the user getting $3$ of his picks to ...
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2answers
5k views

How many bit strings of length 10 contains…

I have a problem on my home work for applied discrete math How many bit strings of length 10 contain A) exactly 4 1s the answer in the book is 210 I solve it $$C(10,4) = \frac{10!}{4!(10-4)!} = ...
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0answers
59 views

Why does $\operatorname{SL}_2(3)$ only yield even permutations?

$\newcommand{\pa}[1]{\left(#1\right)} \newcommand{\pamat}[1]{\left(\begin{array}{cc} #1 \end{array}\right)}$ In an exercise lesson for Algebra 3, we were proven that the special linear groups over ...
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2answers
84 views

Probability of Boys and Girls in Row

Ten male friends and six female friends line up next to the bus stop in a row. Everyone just positions themselves at random. What is the probability that no two females are sitting next to each other? ...
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0answers
23 views

If $g=(n-j,\dots,n)$ and $\sigma\in S_{n-1}$, why are the inversions of $g\sigma$ the union of the inversions of $\sigma$ and $g$?

I can't see why a claim I'm reading is true. If $\sigma\in S_n$, let $R(\sigma)=\{(i,j):i<j,\ \sigma(i)>\sigma(j)\}$, i.e., $R(\sigma)$ is the set of inversions of $\sigma$. The set $C_n=\{1,(...
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2answers
82 views

Number of combinations for N elements with M states

I have $N$ elements: $$e_1, e_2, e_3, e_4, ..., e_N$$ and each of them has $M$ possible states: $$e_i:\, e_{i1}, e_{i2}, ..., e_{iM}$$ I need to find the total number of combinations of these $N$ ...
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1answer
86 views

Probability of an event if the sample space has identical elements

Suppose we have a box, with only one small hole. Suppose 10 distinct black balls and 20 distinct white balls are put in the box. Now, in a random draw of 1 ball, the probability that the ball drawn is ...
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4answers
163 views

Is there any neat way to show $\phi$ is a homomorphism?

In Michael Artin's Algebra (chapter 2, page 50, example 2.5.13) the author illustrates a homomorphism from $S_4$ (all permutations of indices $(1,2,3,4)$) to $S_3$ (all permutations of indices $(1,2,3)...
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3answers
207 views

Rearrangements that never change the value of a sum

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_k = \lim_{n\to\infty} \sum_{k=1}^n a_{f(k)...
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2answers
115 views

$S_6$ 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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1answer
26 views

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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3answers
131 views

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 ...
0
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1answer
27 views

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 ...