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Combinatorics

Set Operations

1. Write the following in roster form: Set N is the set of natural numbers between ten and sixteen 2. Express the following in set builder notation: Z = {2, 3, 4, 5, 6, 7, 8, 9, 10} 3. For sets A and B, determine whether A = B, A is a subset of B, or B is a subset of A

Combinations Expansion Subsets

1. The coefficient of x^7y^2 in the expansion of (2x-y)^9 2. How many subsets of {2,3,5,7,11, 13, 17,19,23) contain four numbers? 3. If a committee varies its meeting days, how many meetings must it schedule before we can guarantee that at least two meetings will be held on the same day of the week?? 4. How many differe

Show that a set of real rational functions is a field.

NOTE: In this description, R represents the symbol for the set of real numbers. I couldn't find a way to type or copy the correct R symbol for the set of real numbers. Also, the parentheses in R(x) is used to distinguish the ring R(x) of rational functions from the ring R[x] of polynomials. Show that the set R(x) of rational

Graphs, Digraphs, Trees and Forests

I am posting one problem from Exercise 2.2, I need answer for 2.9. I am posting another question from Exercise 3.1: Problem 3.2.) Prove that a graph G is a forest if and only if every induced subgraphs of G contains a vertex of degree at most 1. Problem 3.1) Draw all forests of order 6. See attached file for full pr

Combinations and Permutations

1) Identify each of the following 1) as a permutation or combination and 2) as with or without replacement: a) Social security numbers b) Books in your backpack c) Numbers chosen for the "Big" lotto d) The cards in your hand for a card game e) Lunch chosen by a student from the cafeteria menu 2) A

Statistics Problem Set

See attached file for full problem description. 1. If A  B and B  C, what can you conclude? Why? What if A  B and B  C? If A  B and B  C? 2. Write down all possible subsets of {a, b, c, d} 3. Without writing them down what are the number of subsets of the set A = {a, b,

Permutation or Combination Calculations

For each question determine if the situation is a permutation or combination and if it is with or without replacement, then determine the answer. 1 Identify each of the following 1) as a permutation or combination and 2) as with or without replacement: a) Social security numbers b) Books in your backpack c) Numbers chosen

Discrete Math : Combinations, Permutations and Probability (25 MC Problems)

This test consists of 25 equally weighted questions. 1. Given a two-step procedure where there are n1 ways to do Task 1, and n2 ways to do Task 2 after completing Task 1, then there are _________ ways to do the procedure. a. n1 + n2 b. n1 log n2 c. n1 * n2 d. n12 2. How many bit strings of length 10 begin with 1101? a.

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

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

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?

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

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 ≡ 4 (mod 8), your friend Phil Lovett wrote down the following steps: 2x+6 ≡ 4 (mod 8) x+3 ≡ 2 (mod8) x ≡ −1 (mod 8) From here, Phil concludes that the solution set to 2x + 6 ≡ 4 (mod 8) is {x; x ≡ −1 (mod 8)}. (a) Is Phil's

Problem Set

(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

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

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?

Negative Integers Proof : 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..

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