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# Combinatorics

### Isomorphisms, Cyclic Groups and Groups of Permutations

Consider the group Z[4] Ã— Z[6] under * such that (a, b) * (c, d) = (a +[4] c, b +[6] d). (here +[4] means + is in Z[4] and +[6] is in Z[6]) We would like to find a group of permutations that is isomorphic to Z[4]Z[6]. Is this group cyclic? If so, prove it. If not, explain why. Do I need to list all the members and ch

### Combinations

Is it possible to evaluate C(9,12)? Explain.

### Combinations Word Problems

A young boy sends his brother to pick 5 game-boy cartridges from his collection of 10 arcade and 5 sports games. How many ways are there that his brother will select 2 sports and 3 arcade games respectively?

### Combinations : Finding the nth Term

Using standard labeling of a triangle, you are given: B=49.3degree, C=59.6degree, b=4760 Find side a

### Combinations : Finding the nth Term..

Find the 7th term of (x-30)^11

### 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

### Permutation Groups: Another Counting Principle

Modern Algebra Group Theory (CVIII) Permutation Groups Another Counting Principle Using the theorem ' If O(G) = p^n , where p is a prim

### 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

### Counting

1. how many license plates can be made using three letters followed by the three digits or four letters followed by two digits 2. how many bit strings of length 10 contain either 5 consecutive 0s or five consecutive 1s.

### 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

### Proofs with Sets

(See attached file for full problem description). a) Determine the irreducibility of x20-11 over Q(set of rationals), and use it to prove or disprove that the ideal <x20-11> is a maximal ideal of Q[x]. b) Construct an integral domain R and an element a in R such that a is irreducible but not prime in R. c) Suppose that R is

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

### Trees - Counting Problems and Combinations

A Pasta bar has a build your own pasta option. On this option, the menu lists 4 types of pasta noodles, 6 different vegetable choices, 4 meat choices, 3 cheeses and 3 sauces. Each customer who orders this option must choose 1 type of noodle and 1 type of sauce. The customer can choose as many of the vegetables as desired and up

### Permutations for Different Cycles

Some boys come into a room and sit in a circle. Four girls come in and arrange themselves between 2 of the boys, that is , the 4 girls join the circle, but all sit together. If there are 2880 different circles that could be formed in this way, how many boys are there?

### Sampling and Combinations : A store selects four items from a selection of 6 items to arrange in a display. How many different arrangements are possible?

A store selects four items from a selection of 6 items to arrange in a display. How many different arrangements are possible? A. 15 B. 24 C. 360 D. 6

### Sampling and Combinations : A group of people consists of 14 men and some women. One man and one woman can be selected in 252 ways. There are _____ women in the group.

A group of people consists of 14 men and some women. One man and one woman can be selected in 252 ways. There are _____ women in the group. A. 238 B. 18 C. 16 D. 152

### Function Example Problem

F(x) = (mx +7)^1/2 where x >= -7/m and m is a positive constant. It is given that y = f(x) and y = f^-1 (x) do not meet. Explain how it can be deduced that neither curve meets the line y = x, and hence determine the set of possible values of m.

### Satisfy Specific Properties - Bounded Above or Below and Integer

In each part of this problem, display an example of a set M with the specific property or properties. a) M does not equal R (R is the set of all real numbers), but M is bounded neither above nor below. b) M is bounded above but fails to contain its least upper bound. c) M is the set of integers that contains neither a s

### Negative Integers Proof : Denote by -P the set of all negative

2. Denote by -P the set of all negative integers, i.e., the set to which the number m belongs only in case there is a member n of P such that m = -n. If the number m is in -P and Z is a number such that m<Z, then Z is not in P. If the conjecture is true, prove it. If it's false, prove that its false by counterexample or a proo