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Combinatorics

Combinations, Permutations and Cycles

How many elements of order 5 are there in S_8? - I think there are 8! / 3!5! = 56 ways to order the elements in the cycle but how many of order 5 are there? keywords: S8

Combinations

2. A seven-person committee composed of Adam, Betty, Cameron, David, Edward, Fritz, and Grace is to select a chairperson, secretary, and treasurer. How many selections are there where Betty is the chairperson, and Adam and Edward are not officers? 12 20 24 210 None of the above

Permutations and R-Cycles

1. If alpha is an r-cycle, show that alpha^r = (1). [There's a hint that If alpha = (i sub 0 ... i sub r-1), show that alpha ^k(i sub 0) = i sub k.] 2. Show that an r-cycle is an even permutation if and only if r is odd. 3. If alpha is an r-cycle and 1<k<r, is alpha^k an r-cycle?

Different ways: combination and permutation

In how many ways can 7 instructors be assigned to seven sections of a course in mathematics? How many different ways are there for an admissions officer to select a group of 7 college candidates from a group of 19 applicants for an interview? A man has 8 pairs of pants, 5 shirts, and 3 ties. How many different outfits can

Permutations and Combinations..

A) How many ways can a person select three appetizers and two soups if there are six appetizers and five soups on the dinner menu? b) Three married couples have bought tickets for six seats in a row for a movie. i) In how many ways can they be seated? n! = 6 x 5 x 4 x 3 x 2 x 1 = 720 ii) In how many ways can they be seate

Probability, Permutations, Combinations

1. 35% of a store's computers come from factory A and the remainder come from factory B.2% of computers from factory A are defective while 1% of computers from factory B are defective. If one of the store's computers is selected at random, what is the probability that it is not defective and from factory A? 2. Two stores sell

Group Actions and Transitive Permutations

A) Show that if n is odd then the set of all n-cycles consists of two conjugacy classes of equal size in An b) Let G be a transitive permutation group on the finite set A with |A|>1. Show that there is some g in G such that g(a) is not equal to a for all a in A. (Such an element g is called a fixed point free automorphism) c

A television commercial for Little Caesars pizza announced that with the purchase of two pizzas, one would receive free any combination of up to five toppings on each pizza. The commercial shows a young child waiting in line at Little Caesars who calculates that there are 1,048,576 possibilities for the toppings on the two pizzas.

1. A television commercial for Little Caesars pizza announced that with the purchase of two pizzas, one would receive free any combination of up to five toppings on each pizza. The commercial shows a young child waiting in line at Little Caesars who calculates that there are 1,048,576 possibilities for the toppings on the two pi

Permutation Groups and Commutativity

Let Y=(u v/u^4=v^3=1,uv=u^2v^2) Show that a) v^2=v^-1 b) v commutes with u^3 c)u commutes with u d)uv=1 e)show that u=1, deduce that v=1 and conclude that Y=1

Permutations as Product of Disjoint Cycles

Modern Algebra Group Theory (LXXIX) Permutation Groups The Orbits and Cycles of Permutations Write the given permutation as the product of disjoint cycles 1 2 3 4 5 6 6 5 4 3 1 2 The fully formatted problem is in the attached file.

Functions and countable sets

(See attached file for full problem description with all symbols) --- 2.14 (I) Prove that an infinite set X is countable if and only if there is a sequence of all the elements of X which has no repetitions. (II) Prove that every subset S of a countable set X is itself countable. (III) Prove that if

Solutions Modulo congruences

In order to solve the congruence 2x + 6 &#8801; 4 (mod 8), your friend Phil Lovett wrote down the following steps: 2x+6 &#8801; 4 (mod 8) x+3 &#8801; 2 (mod8) x &#8801; &#8722;1 (mod 8) From here, Phil concludes that the solution set to 2x + 6 &#8801; 4 (mod 8) is {x; x &#8801; &#8722;1 (mod 8)}. (a) Is Phil's

Combinations of Integers

Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11 Is the conclusion true if 6 integers are selected instead of 7 How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers a

Sets and Set Operations : Union and Intersection

Create two sets. Set A will be the list of the five items you personally need to buy the most (essential items). Set B will be the list of the five items that you want to buy the most (fun stuff). List the items in Set A and Set B, and also list the items in the unions and intersections of Set A and Set B. Now assume that the p

Combinations for Test Answers

A teacher makes 2 tests that each have: -5 multiple choice questions with 4 possible answers -10 true/false questions (2 possible answers) If a students takes both tests. How many possible combinations of answers are there? Would like to know how to figure this out. Is there a forumula? Does it involve factorials?

Combinations and Permutations

11. A computer lab contains the following computers - a Hewlett Packard, a Compaq, a Sony, a Dell and 3 different models of Macs. How many different ways can the 7 computers be arranged so that the Macs are all together? (You may assume the computers are all in one line.) 12. A public pool employs 17 lifeguards of whic

Counting Problems and Trees

In one residence, cell phones, lap top computers and digital tvs are very popular among students. In fact, all of the students own atleast one of these items, although onlu 15 own all 3. Cell phones are the most popular with twice as many students owning cell phones as own lap tops. and digital tvs are still rare, since only hal

Cardinality, Countability and Denumerable Sets

1. Show that if A and B are countable and disjoint, then A U B is countable. 2. Show that any set, A, of cardinality c contains a subset, B, that is denumerable. 3. Show that the irrational numbers have a cardinality c. 4. Show that if A is equivalent to B and C is equivalent to D, then A x C is equivalent to B x D.

Counting Principles and Probability : Combinations, Is the game fair? and Z-score

In a Chinese restaurant, the menu lists 8 items in Column A and 6 items in column B. To order a dinner, the diner is told to select 3 items from column A and 2 from column B. How many dinners are possible? Suppose a family plans 6 children, and the probability that a particular child is a girl is 1/2. Find the probability

Proofs : Denumerable Sets and Cartesian Products

Need help in determining the following proof exercise. (See attached file for full problem description) --- Corollary 14.2.4: If A is a denumerable set then so is A^n for every positive integer n. Proposition 14.2.3: If A and B are denumerable sets then so is their Cartesian product , AX B. ---

Counting measure

Definition: For any E in X, where X is any set, define M(E) = infinity if E is an infinite set, and let M(E) be then number of points in E if E is finite. M is called the counting measure on X. Let f(x) : R -> [0,infinity) f(j) = { a_j , if j in Z, a if j in RZ} ( Z here is counting numbers, R is set of real numbers)

Lebesgue Measurable Set

Let A be a set in R^n, we denote by A + x_o a parallel shift of A by x_o to A + x_o, A + x_o = { x : x = y + x_o, y in A}. Now, if A is a lebesgue measurable then show that 1). x_o + A is also lebesgue measurable 2). m(A) = m(x_o + A) Can someone check my answer and tell me if it is correct or not? My work: s

Power set and bijection

I have two small problems. I need all the work shown and in the second problem please answer in detail and NOT just yes or no. (See attached file for full problem description)

Combinations Application Word Problem

Suppose that 30 different computer games and 20 different toys are to be distributed among 3 different bags of Christmas presents. The first bag is to have 20 of the computer games. The second bag is to have 15 toys. The third bag is to have 15 presents, any mixture of games and toys. How many ways are there to distribute these

Combination Application Word Problems and Summations

Exercise # 1 A) How many nonnegative integer solutions are there to the pair of equations: And B) How many ways are there to distribute 20 toys to m children such that the first two children get the same number of toys if: 1- The toys are identical? 2- The toys are distinct? Exercise #2