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# Combinatorics

### 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!.

### Permutation and combination

4. Suppose that an unfair coin comes up tails 53.2% of the time. The coin is flipped a total of 14 times. a) What is the probability that you get exactly 7 heads? b) What is the probability that you get exactly 10 tails? c) What is the probability that you get at most 3 heads?

### Permutation and combination

Suppose that you are dealt a hand of 8 cards from a standard deck of 52 playing cards. a) What is the probability that you are dealt no diamonds and exactly 1 jack? b) What is the probability that you are dealt no more than 1 spade? c) What is the probability that you are dealt at least 2 sevens and at least 3 fives?

### How many ways can a disc jockey play ten songs?

A disc jockey has 10 songs to play. Seven are slow songs, and three are fast songs. Each song is to be played only once. In how many ways can the disc jockey play the 10 songs if: - The songs can be played in any order. - The first song must be a slow song and the last song must be a slow song. - The first two songs mus

### 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

### Using the Counting Principle in Probability Theory

A mini license plate for a toy car must consist of a letter followed by two numbers. Each letter must be a C, A or R. Each number must be a 3 or 7. Repetition of digits is permitted. a) Use the counting principle to determine the number of points in the sample space. b) Describe how a tree diagram would represent this si

### Determining the Number of Ways of Getting Marbles

A bag contains three red marbles, three green ones, one lavender one, two yellows, and four orange marbles. How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors?

### 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

### Linear operators

Determine if the following two systems have the same solution set: { -x + y + 4z } { x + 3y + 8z } {(1/2)x + y + (5/2)z = 0} and { x - z = 0 } { y + 3z = 0 } ****Please show detailed work and provide thorough explanation.****

### 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

### Counting is discussed.

Using all 30 runners in each of the following cases, how many different outcomes are possible? We are reporting on the ranking of each runner in the marathon. This is a qualifying marathon for the Olympics and only the fastest eight runners advance. The top three runners will receive gold, silver and bronze medals.

### What is the probability that a randomly picked cube has at least one of its sides painted red?

1. A cube has all of its sides painted in red and then is cut into 216 cubes of the same size. All cubes are placed in an urn and are thoroughly mixed, so that the probability of being randomly picked from the urn is the same for all cubes. What is the probability that a randomly picked cube has at least one of its sides painted

### Preschool Math Differentiated Instruction

What should be included to describe the background of the educational classroom and the goals for counting for preschoolers?

### Equivalence relations and the empty set

Regarding the set of natural numbers N, answer the following questions. Is â?? a binary relation? Explain. Is â?? reflexive? Explain. Is â?? symmetric? Explain. Is â?? transitive? Explain. Is â?? an equivalence relation? Explain.

### 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}

### Permutations and Combinations Problems

An electronics store receives a shipment of 30 graphing calculators, including 3 that are defective. Four calculators are selected to be sent to a local high school. A. How many selections can be made using the original shipment? B. How many of these selections will contain no defective calculators?

### Set-Builder Notation to Roster Form

Determine whether A = B, A â?? B, B â?? A, A â?? B, B â?? A or if none of these answers applies. (There may be more than one answer) A = {x | x is a sport that uses a ball} B = {basketball, soccer, tennis}

### Intersection

Given the following sets: A: { a,b,c,d} B: { b, c, e, f} C: {x,y,z) What is the intersection of A and B and C? What is the union of A and B and C? Please explain, how you got the answer.

### Inclusion-Exclusion in the English Alphabet

How many permutations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird? Please show all the steps along with necessary explanations.

### Pairwise disjoint sets

Find the number of elements in A1 UNION A2 UNION A3 if there are 100 elements in A1, 1000 in A2, and 10000 in A3 if a) A1 is subset of A2 and A2 is subset of A3. b) The sets are pairwise disjoint.

### Conjugacy classes: symmetric and alternating groups

The conjugacy classes in Sn correspond to partitions of n, and are determined by cycle structure. Which conjugacy classes of Sn that are in An split into two conjugacy classes in An? Please give a necessary and sufficient condition. Sn is the symmetric group of degree n, and An is the alternating group of degree n.

### Permutations and Combinations of Strengths

12. How many bit strings of length 12 contain a) exactly three 1s? b) at most three 1 s? c) at least three 1 s? d) an equal number of 0s and 1s?

### Counting for Wedding Photography

In how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, where the bride and the groom are among these 10 people, if a) the bride must be in the picture? b) both the bride and groom must be in the picture? c) exactly one of the bride and the groom is in the the picture?

### Universal Set

Explain what is meant by a universal set. Give a real world example of two universal sets. Be sure to first clearly define the two sets. Describe what is meant by the intersection of two sets and give an example from the previously two created sets of their intersection. Describe what is meant by the union of two sets and

### Counting Principle and Tree Diagrams

A mini license plate for a toy car must consist of three numbers followed by a letter. Each number must be a 3,6, or 9. Repetition of digits is NOT permitted. Each letter must be an A, B, or C. *Use the counting principle to determine the number of points in the sample space. *construct a tree diagram to represent situation

### Sets and Counting

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. Create Set C, by taking the complement of Set B in Set A, i.e. all of the non-essential items in your wallet or