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# Combinatorics

### permutation group

Find a group G and a subgroup H such that for x,xâ??,y,yâ?? in G, xH = xâ??H and yH = yâ??H, yet xyH does not equal xâ??yâ??H. (alternatively find an example of G and H where (xH)(yH) does not equal xyH

### Organized Counting/Combinations

How many subsets of {1, 1, 2, 2, 2, 3, 4, 4, 5} will contain a 2? A group organizing Smith College's alumni reunion includes 6 current students, 8 current staff members and 11 alumni. Two individuals from each category will work on the publicity committee. Use combinations to determine the number of different publicity commit

### Organized Counting & Combinations Practice Examples

What is the value of C(15,4) is: a) 1 365 b) 36 036 c) 3 003 d) 32 760 The number of combinations of 9 items taken 4 at a time is: a) C(4,9) b) C(9,4) c) P(9,4) d) C(9,3) For a Data Management quiz, the teacher will choose 11 questions from the 15 in a set of review exercises. How many different

### Permutations for John's Company

Andrew owns a small Internet company and offers his clients three different formats for the name part of their e-mail address and seven different domains for the second part of the address. In how many ways can a person set up an e-mail address through John's company Part 2 While shopping for a new outfit, Shareen finds fi

### Combinations of lottery drawing

There are 24 numbers with at least 3 or 4 of them guaranteed in a lottery drawing. write down all possible combinations in the 24 number matrix. the numbers are: 02 07 16 35 40 42 15 17 20 23 38 46 03 05 06 08 19 39 07 12 17 20 28 33

### Subset and proper subset of a set

As we know, a set is just a collection of objects that are similar in some way, like a gaggle of geese, pride of lions, or an army of ants. To further classify our world, we can create subsets from a larger set. Let's consider the idea of a subset for our discussion. Give an example of a subset and a proper subset. Explai

### Evaluating sets and subsets

Give an example of a set S such that: a) S is contained in P(N) N=natural #'s b) S is an element in P(N) c) S is containd in P(N) and |S| = 5 d) S is an element in P(N) and |S| = 5 the P is the Power Set

### Fundamental Counting Principles: Permutation & Combination

1.) A club with 10 members is to pick a president, vice president, treasurer, and secretary. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled? 2.) Find the value of the following expression: a. 12P4 b. 12C4 c. 4P4 d.

### Counting Principles and Varieties

28. Automobile Manufacturing An automobile manufacturer produces 7 models, each available in 6 different exterior colors, with 4 different upholstery fabrics and 5 interior colors. How many varieties of automobile are available? 54. Barbie A controversy arose in 1992 over the Teen Talk Barbie doll, each of which was p

### Solve the Equations, Permutations and Combinations

1. As a business owner, there are many decisions that you need to make on a daily basis, such as ensuring you reach the highest production levels possible with your company's products. Your company produces two models of bicycles: Model A and Model B. * Model A takes 2 hours to assemble. * Model B takes 3 hours to assemble.

### Put into practice what you have learned about sets and counting. I want you to create two sets, set A and set B.

Set A will be a list of five items you personally NEED to buy the most (essential items). Set B will be a list of five items that you WANT to buy the most (fun stuff). List the items in Set A and Set B, and also list or state the items in the union and in the intersection of Set A and Set B. Now assume that the prices of th

### Finding the Identity Element Under Addition: Example Problem

What is the identity element of M(3,2,W) under addition? M(m,n,S) is the notation for the set of matrices with m rows and n columns, such that each element is a member of the set S. So M(3,2,W) is the set of 3 x 2 matrices with whole numbers for each element. Thanks.

### Number of Subsets of a Set and Intersection/Union of Sets

1. Without writing them all out, what is the number of subsets of set A ={king, queen, knight, prince, princess, duke}? 2. Given these elements of sets A, B, and C list the elements of set D. Show your work step by step. A = {1, 2, 3, 4} B = {3, 4, 5, 6, 7} C = {3, 5, 7, 9} D = A intersected with (B U C)

### Identify Elements and Create Sets: Example Problem

For the first SLP I want you to put into practice what you have learned about sets and counting. I want you to create three sets, set A, set B, and set C by going through the items in your wallet or purse. Set A will be a list of all of these items. Create Set B, from the items in Set A that you think are essential.

### Set Definitions with Unions and Intersections

Exercises on Sets 1. Please provide a short definition of the following: a. Set b. Subset c. Proper Subset d. Complement of a set e. Union of a set f. Intersection of a set Solve following problems showing your work: 2. Set X = {3, 7, 11, 21, 39, 43, 567}, Set Y = {1, 3, 6, 8, 11, 42, 567} a. What is

### Surjection from P(N) to Least Uncountable Ordinal

Prove that there exists a subjection from P(N) onto omega_1, where N is the set of all natural numbers, P(N) is the power set of N, and omega_1 is the least uncountable ordinal. See attachments for fully formulated problem.

### Similarities and Differences Between Permutation and Combination

What are some similarities and differences between permutations and combinations? Give a real-life example of either a permutation or combination.

### Running Time of a Set Union Implementation

The input consists of two arrays each representing a set of integers (in each array, each value appears only once). The output is an array representing the union of the two sets - again, each value appears only once. Write a method to implement set union and analyze its running time.

### Sets and subsets

For the first SLP I want you to put into practice what you have learned about sets and counting. I want you to create three sets, set A, set B, and set C by going through the items in your wallet or purse. Set A will be a list of all of these items. Create Set B, from the items in Set A that you think are essential.

### Exercises on Sets

Exercises on Sets 1. Please provide a short definition of the following: a. Set b. Subset c. Proper Subset d. Compliment of a set e. Union of a set f. Intersection of a set Solve following problems showing your work: 2. Set X = {3, 7, 11, 21, 39, 43, 567}, Set Y = {1, 3, 6, 8, 11, 42, 567}

### Combinatorial problems with counting principles

1. Use the theorem below to determine the number of nonequivalent colorings of the corners of a rectangle that is not a square with the colors red and blue. Do the same with p colors. (the answer is (p^4+3p^2)/4...i just dont know how to get there) Theorem: Let G be a group of permutations of X and let C be a set of color

### Multiplication Principle, Permutations, Combinations, Etc.

Please provide the formulas and 2 solved examples using the formulas for each of these topics: The Multiplication Principle Permutations Combinations Probability Applications of Counting Principles.

### Probability distributions, probability, combinations, permutations

Mat 102 1. How many different 5-digit sequences can be formed using the digits 0,1,....8 if repetition of digits is allowed? Use the multiplication principle to solve the problem 2. A shirt company has 4 designs each of which can be made with short or long sleeves. There are 7 color patterns available. How many different

### Probability

A mini license plate for a toy car must consist of a one digit odd number followed by two letters. Each letter must be a J or K. Repetition of letters is permitted. Use the counting principle to determine the number of points in the sample space. Construct a tree diagram to represent this situation and submit it to the w

### Permutations for Selecting Representatives

A representative is to be selected from each of 3 departments in a small college. There are 7 people in the first department, 5 in the second department, and 4 in the third department. a. How many different groups of 3 representatives are possible? 3360 b. How many groups are possible if any number (at least 1) up to 3 rep

### Generating Permutations and Combinations

A.Consider the partial order less than or equal to(<=) on the set X of positive integers given by "is a divsor of." Let a and b be two integers. Let c be the largest integer such that c<=a and c<=b and let d be the smallest integer such that a<=d and b<=d. What are c and d? b. Prove that the intersection of R and S of two equ

### Math

Create three sets, set A, set B, and set C by going through the items in a wallet or purse. Set A will be a list of all of these items. Create Set B, from the items in Set A that you think are essential. Create Set C, by taking the complement of Set B in Set A, i.e. all of the non-essential items in a wallet or pu

### Generating Permutations and Combinations

The Complement A of an r-subset A of {1,2...,n} is the (n-r)-subset of {1,2,...,n} consisting of all those elements that do not belong to A. Let M= C(n,r), the number of r subsets and at the same time the number of (n-r)-subsets of {1,2...,n}. Prove that if A1,A2,A3...AM are the r subsets in lexigraphic order then complements Am

### total number of possibilities

1. in how many ways can five identical rooks be placed on the squares of an 8 by 8 board so that four of them forma the corners of a rectangle with sides parallel to the sides of the board. 2. consider and 9 by 9 board and nine rooks of which five are red and four are blue. suppose you place the rooks on the board in nonattac

### Probability

1. Four (standard) dice (cubes with 1,2,3,4,5,6 respectively,dots on their six faces) each of a different color are tossed each landing with one of its faces up, thereby showing a number of dots. Determine the following probabilities. a. the probability that the total number of dots shown is 6 b. the probability that at most