Questions tagged [permutations]

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

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-2
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6 views

4 digit code probability

You are creating a 4 digit pin code. how many choices are there in the following cases: no digits are repeated no restriction no digits repeated but 7 and 8 are present
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How do i solve the following questions in the picture about combinations.

Picture of question I don't know how to go about to solve the questions
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1answer
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Combinatorics Question with vowels

How many 8-letter words contain exactly 5 vowels (a,e,i,o,u)? What if repeated letters were not allowed? This question has two parts to be answered. The first part is,"How many 8-letter words ...
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Characterizing Special Permutations

Consider the multiset $$\mathcal{S} = \{1,1, 2,2, \ldots, n,n\}$$ Let $A_1 A_2 \ldots A_{2n}$ be some permutation of $\mathcal{S}$. Now we want to eliminate all permutations where $A_i = A_{i + 1}$, $...
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Find the optimal permutation

Given $n$ numbers $a_1 < a_2 < \cdots < a_n$, find a permutation $\sigma$ such that $$\sum\limits_{1\leq i\leq n-1} \exp(\sigma(a_i)+\sigma(a_{i+1}))$$ is the largest.
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2answers
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Calculation of finding way to make a password

A computer password requires you to use exactly 1 uppercase, exactly 3 lowercases, 3 digits and 2 special charecters(given that there are 33 special charecters that can be used)(every thing can be ...
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1answer
144 views

Special permutations of $\{1,2,3,\ldots,n\}$

How do you show that number of permutations of $\{1,2,3,\ldots,n\}$ such that image of no two consecutive numbers is consecutive is $$n! + \sum_{k = 1}^{n}(-1)^k\sum_{i = 1}^{k}\dbinom{k - 1}{i - 1}\...
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4answers
39 views

Number of functions with a fixed number of elements in the range

I have a question that goes: Find the number of functions $f:A \longrightarrow B$, such that the range contains exactly $3$ elements. Given that $n(A)= 4$ and $n(B)= 5$ Here's what I tried. Range has ...
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1answer
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$X$th Permutations of $1,2,3,4,5$ is $25314$. Find X

The permutations of $1,2,3,4,5$ are lexicographically ordered. $X$th permutation is $25314$. Find $X$. I am getting $1*4! + 3*3! + 1*2! + 1=45$. Is it correct? Reasoning: There are $1*4!$ numbers of ...
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condition when a twisted wreath product is quasiprimitive

I am going through the definition of a twisted wreath product defined in Dixon and Mortimer's book Permutation groups, which goes as follows. Let $T$ and $K$ be arbitrary groups and let $L \leq K$ ...
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2answers
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If Anna goes from point A to point B, each step can only move up or move right. How many method(s) is / are there?(reference the grid below)

If Anna goes from point A to point B, each step can only move up or move right. How many method(s) is / are there?(reference the grid below) I’ve just recently learned permutations and combinations. ...
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3answers
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Question regarding number of non decreasing functions

I have a question that requires me to find the number of non decreasing functions $f: A \longrightarrow B$ where $A=\{1,2,3,4,5\}$ and $B = \{-2,-1,0,1,2,3,4,5\}$ I tried doing this by finding the ...
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Permutation of sequences

Suppose $X$ is the sequence of zeroes and ones of length $m$. Let $x_i\in \{0,1\}$ for $1\leq i \leq m$. $$X=\{x_1,x_2,\ldots, x_m\}$$ Similarly let $Y$ be such a sequence of length $m$ such that the ...
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3answers
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Where is my error in this combinations word problem?

A student has to answer $10$ out of $13$ questions in an examination. The number of ways in which he can answer if he must answer at least $3$ of the first five questions is: The answer is not $^5C_3 ...
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2answers
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Find $\binom{80}{40}\bmod 2000$ [duplicate]

Find $\displaystyle\binom{80}{40}\bmod 2000$. So far, I've found that $\displaystyle\binom{80}{40}$ is divisible by $2^2$ and $5^1$, so the answer isn't $0$. Usually, with smaller numbers, I would ...
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2answers
28 views

Permutations with No letters-digits repetitions

A car license plate consists of 3 capital letters of the English alphabet in the first 3 positions of the license plate followed by 4 digits from 0 to 9. How many different plates can we have if ...
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Double sums over permutations

I'm sorry maybe it's obvious by why is there a p! which appear in the right-hand side? And have we used a change of variable like $ \sigma '$ = $\sigma \circ \gamma$ ? $\left(\frac{1}{p !}\right)^{2} ...
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1answer
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How to find the number of onto and into functions or increasing/decreasing functions, given certain conditions?

I have two related questions that I have a problem in. Here's the 1st one: What I mean by into functions: Function f from set A to set B is Into function if at least set B has a element which is not ...
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Probability for drawing balls in a monotonic (increasing and decreasing) order with order and without replacement

Having an urn with 7 balls each having printed its numbers 1 - 7 on it, we draw 4 of them without replacement and order matters. I am trying to find the probability of drawing them in a monotonic ...
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When is longest increasing subsequence at least N/2 long?

When analizing some algorithms I came up with this problem. Suppose we have a sequence a with N elements. The goal is to find (aproximately) the ratio of #{permutations of a s.t. the LIS is at least N/...
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1answer
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Conjugacy Class of $(2 1)$ in $S_4$

What are the conjugacy classes of $(21)$ in $S_4$? I think they are $\{(12),(13),(14),(23),(24),(34)\}$ But I'm not sure and the centralizer of $(12)$ is $\{(21),(34)\}$. Is this right?
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Variations of the problem of the number of ways to divide $N$ items into $K$ bins

I can use the stars and bars approach to solve the problem of dividing $N$ items into $K$ distinct bins with no restrict on the number of items each bin has to have. It's simply $$ \binom{N+K - 1}{K - ...
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2answers
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Dividing $7$ balls into $5$ baskets and finding the probability each basket has at least 1 ball

We are given $7$ balls and $5$ baskets. We want to place each ball into a basket and the probability of each ball assigned to a specific basket is independent of each other. How do we find the ...
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1answer
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Express a problem as function/sets

It has been a while since I last practiced writing functions with appropriate notations. I want to summarize this problem statement through functions/any other mathematical expression. Basically, I ...
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1answer
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2 numbers are chosen from $1,3,5..151$ and multiplied in all possible ways. Find number of ways such that product is a multiple of 5

Number of available elements = 76 Elements that are a multiple of 5 = 15 Now one perfectly acceptable answer for this is $$\binom {30}{2} +\binom{15}{1}\times \binom {46}{1}$$ I understood how it came,...
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1answer
28 views

Find the total number of positive integral solutions for $x,y,z$ such that $xyz=24$

I recognize that this is related to the stars and bars algorithm, but I am just not able to apply it here The general method is factor the number, in this case is $2^3\times 3$, but what next? It’s ...
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1answer
46 views

Correctly serving drinks in a cafe problem

Five friends go into a cafe, sit at a table and order 3 coffees and 2 teas. The waiter brings the drinks to the table and randomly serves the drinks to the five friends. How do I compute the ...
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30 views

Number of ways to arrange people in straight line and round table.

I have the solution to 2 counting questions. But I do not understand why in Q1, it is $2!$ but in Q2 it is $(2!)^5$. These steps in my understanding takes account for order within each "group&...
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Establish formula for a combinatoric problem

This is a hard problem. Please read the following. There can be multiple farms and multiple farmers in a problem. Each farmer can attend to either one farm or two adjacent farms at a time- whether a ...
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1answer
36 views

How many length 10 words can we make that start with “TRY” or end with “TRY”?

With replacement and repetition is allowed, if we can choose from the 26 letters of the alphabets, how many length 10 strings/ words can we make that either start with the substring "try" or ...
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Permutation with exactly one pair of inverse order [duplicate]

I came up with an elementary counting problem for which I could find a simple solution. Please help. N people of different heights line up single-file, in how many ways there is exactly one person ...
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1answer
41 views

$10$ identical black balls, $5$ identical red balls, $2$ identical white balls. What is the probability of choosing three black balls from this box?

I have $10$ identical black balls, $5$ identical red balls, $2$ identical white balls in my box. What is the probability of choosing three black balls from this box? My Attempt: We have to first ...
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1answer
44 views

How to distribute 10 distinct envelopes into 3 (identical) mailboxes?

As in the title, I would like to distribute 10 distinct envelopes into 3 mailboxes. Let us also assume that some mailboxes can remain empty. If they were identical envelopes, I know that we can use ...
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1answer
20 views

How could the math for this combinatorics solution be simply explained?

I am faced with organizing a team of 5 speakers: $A,B,C,D,E$. I must figure out how many permutations I can organize the five speakers, such that $Speaker\,A$ presents before $Speaker\,B$. A valid ...
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2answers
52 views

The students $S_1, S_2,…S_{10}$ are divided into 3 groups A, B and C

It can be seen as distributing n unique objects above m groups The students $S_1, S_2,...S_{10}$ are divided into 3 groups A, B and C such that each group has at least one students and C has at most ...
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1answer
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Can these connections be considered as permutations or combinations?

Any help would be highly appreciated.
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Why is this problem a question on the topic of permutations/combinations?

I am reviewing some random problems about combinations/permutations. I stumbled upon the following: In a group of $20$ people, how long will it take each person to shake hands with each of the other ...
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2answers
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Confusion in one of the combinatorics/probability problem, with my approach for its solution.

The only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk ...
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1answer
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If $\beta = \left( 1 \;3 \;5 \;7 \;9 \;8 \;6\right)\left(2\; 4 \;10\right) \in S_{10}$, what is the smallest integer such that $\beta^n = \beta^{-5}$?

I manually found that $\beta^{-5} = \left(1\; 5 \;9 \;6 \;3 \;7 \;8\right)\left( 2 \;10 \;4\right)$. The nice thing is that since beta is composed of two disjoint cycles, so I know that $\beta^n = \...
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1answer
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How do I permute a list of integers so that as many possible integers will move to a place where a smaller integer used to stand?

I'm trying to understand the solution to an algorithm/logic problem. The problem statement is as follows: You are given an array of integers. Vasya can permute (change order) its integers. He wants ...
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0answers
19 views

Find The Number of ways in which letters of the word ENGINEER can be arranged so that no two alike letters are together [duplicate]

My attempt was based on principles of exclusion-inclusion but I'm unsure how to exclude in this case as the arrangements in which 'EEE' occur together are a complete subset of the arrangements in ...
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1answer
31 views

How many elements associated (in the same conjugacy class) with a $\sigma \in S_{3n}$

Let $\sigma = (123)...(3n-2, 3n-1, 3n) \in S_{3n}$ The question is how many elements associated (in the same conjugacy class) with a $\sigma \in S_{3n}$ As I understood we can see how transposition ...
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0answers
62 views

Unitary transformation equivalent to permutation

Suppose we have a real, symmetric, positive semi-definite matrix $A$ which we flatten into a vector $\mathrm{vec} \left[ A \right]$. I am interested in solutions to the following equations, $P \ \...
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Is this way correct to solve this question in Permutations and Combinations?

I have this question in Permutations and Combinations, You have $3$ sections $A, B$ and $C$. You need to answer $12$ questions out of $20$ in section $A$. $4$ out of $6$ from section $B$. In section $...
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2answers
50 views

Left coset and right coset of subgroup $H=\langle(234)\rangle$ in alternating group $A_4$

My homework question is: Partition $G=A_4$ into left cosets of the subgroup $H=\langle (234)\rangle$ but I am not sure how to start with. I know that $$A_4= \{(1), (12)(34), (13)(24), (14)(23), (123)...
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2answers
58 views

Probability that position $i$ is a peak in $\sigma$, where $\sigma$ be a uniformly random permutation of $\{1,\ldots,n\}$

Let $\sigma$ be a uniformly random permutation of $\{1,\ldots,n\}$. That is $\sigma(1),\sigma(2),\ldots, \sigma(n)$ is a permutation and it is chosen uniformly from one of the $n!$ permutations. ...
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4answers
40 views

Probability of 4 colors sequence with repetition (8 choices of color)

Knowing that the sequence contains exactly two colors, what is the probability that one of the two colors will be repeated exactly three times? A = The probability that one of the two colors will be ...
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2answers
47 views

Expected discount shop owner will have to shell out in a guessing game with customers

This question is a bit of a corollary to this one: Generate a random permutation of the first $n$ numbers on the fly I have some customers coming to my shop. I'll accept the first $n$ and then close ...
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0answers
100 views

Generate a random permutation of the first $n$ numbers on the fly

I have a large number of people coming to my shop and want to assign ids from $1$ to at-most $n$ to all of them. While I don't know in advance how many people will arrive, I'll simply stop accepting ...
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1answer
33 views

Subset permutations

Given that $B = (a_1, a_2 \cdots, a_{12})$ is a permutation of the set $(1, 2, \cdots, 12)$ such that $a_1>a_2>a_3>a_4>a_5>a_6$ and $a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{...

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