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Combinatorics is a sub-discipline of Algebra which is concerned with the study of combination, enumeration and permutation of sets of elements. One of the main aspects of combinatorics is the study of combinatorial structures found in an algebraic context. It draws on principles from Group Theory and Representation Theory.

Representation Theory is the study of abstract algebraic structures. It aims to represent their elements in the form of linear transformations of vector spaces; in essence, representation makes an abstract mathematical concept more concrete. Thus, it can be seen that this particular theory is especially prominent in the study of Combinatorics – as it hints at the appearance of enumerative methods in the area of algebraic geometry.

Algebraic Combinatorial problems are not limited to the discipline of algebra, but rather they can extend into the realm of probability, geometry and topology. In fact, this study can go even further, as it has many applications in the fields of optimization, computer science and physics. Thus, understanding Combinatorics is crucial for the study of Algebra as well as other disciplines which examines combination, enumeration and permutation.

Democrats and Republicans on Task Force

A certain city has experienced a relatively rapid increase in traffic congestion in recent times. The mayor has decided that it's time to do something about the problem and decides to form a task force to research the causes of the problem and come up with suggestions for dealing with it. The mayor first compiles a list of names

Permutations self study

2) The National association of college and university business officers researched the change in university and college endowments from 2007 to 2008.The table below shows the findings of 203 colleges and universities with 2008 endowment levels above 300 million dollar . It indicates how many schools with an endowment level had t

Gaussian Elimination Count

Please help with solving the following question regarding Gaussian eliminations to a system. The system [see the attachment for the matrix and equation] Where a_ij = 0 whenever i-j >= 2 Do an operation count of MD (Multiplications/Division) and AS (Addition/Subtraction) when using Gaussian elimination to solve the syst

Upper bounds on states in the game of chess.

A step by step analysis is given for the following questions, 1. Find an upper bound for the number of possible states in the game of chess, assuming that draw-by-repetition is enforced if the same position is repeated three times. 2. Find an upper bound for the number of possible moves in a single turn in the game of ches

Combinations and Different Ways for Statistical Measurement

I have attempted part of this practice exercise in the text, but I need help in understanding the rest of it. I thought maybe another perspective would enlighten me. Thanks! There are 7 women and 9 men on a faculty. a.) How many ways are there to select a committee of 5 members if at least 1 member must be a woman? b.) How

Basic Permutation Example

Please help with the following combinatorics problems. Sarah, Jolly and Betty are female triplets. They and their 10 cousins are posing for a series of photographs. One pose involves all 13 children. How many ways can the 7 boys and 6 girls be arranged in one row under each of the following conditions? a) The boys and gir

Combinations and Permutations Problem Set

Question 1: From a total of 15 people, three committees consisting of 3, 4, and 5 people, respectively, are to be chosen. b) How many such sets of committees are possible if there is no restriction on the number of committees on which a person may serve? Question 2: c) In how many ways can we select a set of five micro

Sets: Definitions and Examples

Explain and give examples for the following: - Equivalent sets vs. Equal sets - Cardinality of sets and how cardinality relates to the number of subsets of a set - Subset vs. Proper Subset - Complement of a set vs. Complement of a universal set.

Permutations and Combinations-Single Row, Multiple Row Problems

I am studying perms and combs on my own for a course which i want to take in the future. I have come across 2 questions, to which I need a solution. I would also need the reasoning applied to the solution for future application. Here are the 2 questions. Question 1. How many ways can 18 different vehicles be arranged in a c

Discrete Math Counting

How many permutations of the letters ABCDEFGH contain a. the string ED? b. the string CDE? c. the strings BA and FGH? d. the strings AB, DE and GH? e. the strings CAB and BED? f. the strings BCA and ABF? How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other

Dealing cards, counting bagels, permutations of a string

How many ways are there to deal hands of seven cards to each of five players from a standard deck of 52 cards? A bagel show has onion, poppy seed, egg, salty, pumpernickel, sesame seed, raisin, and plain bagels. How many ways are there to choose: a) six bagels? b) a dozen bagels? c) two dozen bagels? d) a dozen bagels wi

Truth Tables of Seven Teachers and Three Children

1) How would you disprove this statement? [P and Q] => [P and R] 2) How many multiples of 4 are there in {n; 37< n <1001} ? 3) Seven teachers and three children are to stand in line for a photo session. a) How many arrangements are there? b) How many arrangements are there if only any 4 teachers and no children are in th

Discrete math lists: MP3 player playlist, photos, rooks

I want to create two play lists on my MP3 player from my collection of 500 songs. One playlist is titled "Exercise" for listening in the gym and the other is titled "Relaxing" for leisure time at home. I want 20 different songs on each of these lists. In how many ways can I load songs onto my MP3 player if I allow a song to

Cardinality, countability

2-12 through 2-14. Exercise 2-12. Consider the integers in the arrangement 0, 1, -1, 2, -2, 3, -3, ... Let n E N. Which integer occupies the 2n position? The 2n+ 1 position? Prove that I and N can be put into one-to-one correspondence. From Exercise 2-12, I has N0 integers. As we have seen, adding one new members to a cou

Abstract Algebra: Burnside's Counting Theorem

Please see the attached file for full problem description. A bead is placed at each of the six vertices of a regular hexagon, and each bead is to be painted either red or blue, how many distinguishable patterns are there under equivalence relative to the group of rotations of the hexagon? Repeat Problem 8 with a regular h

Determining Amount of Group Combinations

A computer programming team has 12 members. (a) How many ways can a group of 7 (out of the 12 members) be chosen to work on a special project? Explain. (b) Suppose that out of the 12 members on the team 7 are IT majors and 5 are math majors. (i) How many groups of 7 people can be formed that contain 4 IT majo

Factorials, combinations and permutations.

Calculating factorials, combinations, permutations. Evaluate the given expressions and express all results using the usual format for writing numbers (instead of scientific notation). Factorial Find the number of different ways that the nine players on a baseball team can line up for the National Anthem by evaluating 9!.

Compare improper and proper subsets.

Sets (Part I) 1. List all the subsets of { 8, 16, 27, 31, 60} 2. Determine the number of subsets of {mom, dad, son, daughter} 3. At MegaSalad, a salad can be ordered with some, all, or none of the following set of ingredients on top of the salad greens: {ham, turkey, chicken, tomato, feta cheese, cheddar chees

Problems in Calculus, Combinatorics and Functional Analysis

Problem 1: A tank contains 100 gal of brine may by dissolving 80 lb of salt in water. Pure water runs into the tank at the rate of 4gal/min and the mixture, which is kept uniform by stirring, runs out at the same rate. Find the amount y(t) of salt in the tank at any time t. Problem 2: For a continuous and onto funct

This problem applies combinations to playing Powerball.

In the multistate lottery game Powerball, there are 120,526,770 possible number combinations, only one of which is the grand prize winner. The cost of a single ticket (one number combination) is $1. Suppose that a very wealthy person decides to buy tickets for every possible number combination to be assured of winning a $150 mil

Analyze probability and logic.

Provide a linear description of all attempts you tried, including those that didn't work. Show all work that represents your process. Also, include the level of challenge you encountered. Was this a problem or an exercise and why? Provide real-life connections to each of the problems you solved. In addition, think about answerin

arithmetic progression method

See two attachments. Each point on the plane will be counted sooner or later, starting from the origin and following the "path" suggested in the figure. Note that the counting starts from 0 not 1 (which is not a big deal). For example, the origin, (0,0), will be counted first. When you plug (0,0) into the formula, you

Defining Sets and Set Operations

Note: U = union, n = intersect 1. Let U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z} C = {v, w, x, y, z}. Determine A U (B n C) 2. Let U = {q, r, s, t, u, v, w, x, y, z} A = {q, s, u, w, y} B = {q, s, y, z}

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

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.