50 Counting Questions
|
|
- Maurice Stafford
- 6 years ago
- Views:
Transcription
1 50 Counting Questions Prob-Stats (Math 3350) Fall 2012 Formulas and Notation Permutations: P (n, k) = n!, the number of ordered ways to permute n objects into (n k)! k bins. Combinations: ( ) n k = n!, the number of unordered ways to arrange n objects into k!(n k)! k bins. This notation also refers to the binomial coefficients and is read n choose k. Cardinality Operator: n(s), counts the (possibly infinite) number elements in set S. We will use word to refer to all possible distinct orderings of a group of letters without concern for whether or not these orderings would show up in a dictionary. Problems 1. There are 16 students in the Math club. How many different ways could they select a president, vice president and treasurer for the club? 2. A website requires the user to choose a password with 5 letters and 2 numbers in that order. Each letter or digit may be used only once. How many different passwords are possible? How does this change if 5 letters and 2 numbers may be used in any order? 3. The name of 11 students are placed in a hat. A teacher will reach into the hat and select 3 names at one time. Each of those 3 students will win the same prize. How many different groups of 3 winners could be chosen? 4. In how many ways can a sorority of 20 members select a president, vice president and treasury, assuming that the same person cannot hold more than one office? 5. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? 6. A license plate is to consist of three letters followed by two digits. How many different license plates are possible if the first letter must be a vowel (a, e, i, o, u), and repetition of letters is not permitted, but repetition of digits is permitted? 7. A teacher is making a multiple-choice quiz. She wants to give each student the same questions, but have each student s questions appear in a different order. If there are 1
2 twenty-seven students in the class, what is the least number of questions the quiz must contain? 8. How many 10 digit numbers can be formed using 3 and 7 only? 9. Everyone shakes hands with everyone else in a room. Total number of handshakes is 66. What is the total number of people in the room? 10. In a college football training session, the defensive coordinator needs to have 10 players standing in a row. Among these 10 players, there are 1 freshmen, 2 sophomores, 4 juniors, and 3 seniors, respectively. How many different ways can they be arranged in a row if only their class level will be distinguished? 11. In one year, three awards (research, teaching, and service) will be given for a class of 25 graduate students in a statistics department. If each student can receive at most one award, how many possible selections are there? 12. A president and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if: 13. A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer? 14. In how many different ways can a true-false test consisting of 8 questions be answered? 15. Find the number of ways to arrange 16 items in groups of 4 at a time (order matters). 16. Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter? 17. How many ways are there to select a subcommittee of 7 members from among a committee of 17? 18. How many ways are there to have a license plate with 3 letters followed by 4 numbers in which the 4 numbers are listed from least to greatest (examples: 1234 or 0569)? 19. How many full house combinations are there in 5-card poker? 20. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men sit on the committee. In how many ways can it be done? 21. In how many different ways can the letters of the word LEADING be arranged in such a way that the vowels are together? 22. Ten people go to a party. How many different ways can they be seated at a round table? 23. There are 10 marbles in a bag numbered from 1 to 10. Three marbles are selected at random without replacement. How many different ways are there of selecting the three
3 marbles? 24. How many ways can I give 3 tin cans to 8 people if each person receives at most one can? 25. An identification code is to consist of seven letters followed by three digits. How many different codes are possible if repetition is permitted? 26. An 8-bit binary word is a sequence of 8 digits, of which each may be either a 0 or a 1. How many different 8-bit words are there? 27. Jane and Thomas are among the 8 people from which a committee of 4 people is to be selected. How many different possible committees of 4 people can be selected from these 8 people if at least one of either Jane or Thomas is to be selected? 28. Georgia license plates consist of three letters followed four numbers. How many license plate combinations are possible if the license plate contains no vowels (A, E, I, O, U, or Y) nor contains the number A company issues a questionnaire whereby each employee must rank the 5 items with which he or she is most satisfied. The items are wages, work environment, vacation time, job security, supervisors, health insurance, break time, and retirement plan. The ranking is to be indicated by the numbers 1, 2, 3, 4 and 5, where 1 indicates the item involving the greatest satisfaction and 5 the least. In how many ways can an employee answer this questionnaire? 30. A keypad lock has 10 different digits, and a sequence of 5 different digits must be selected for the lock to open. How many key pad combinations are possible? 31. There are five women and six men in a group. From this group a committee of 4 is to be chosen. How many different ways can a committee be formed that contain three women and one man? 32. How many 1 pair combinations are there in a standard deck of 52 cards? 33. A class of 30 students enter a room with 40 desks. How many different ways can those students be seated? 34. If a school has lockers with 50 numbers on each combination lock, how many possible combinations using three numbers are there? 35. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are on the committee. In how many ways can it be done? 36. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? 37. There are 6 boxes numbered 1, 2, Each box is to be filled up either with a red or a green marble in such a way that at least 1 box contains a green marble and the
4 boxes containing green marbles are consecutively numbered. The total number of ways in which this can be done is? 38. How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed? 39. If a license plate has exactly 3 letters and 4 digits, how many license plates are possible if there must be exactly one vowel (excluding Y) and one 7 in the sequence? 40. Proteins are made up of chains of amino acids. Insulin is a relatively small protein with 53 amino acid residues. How many possible proteins of length 53 can be made with 20 possible amino acids for each position in the protein? 41. We wish to know how many color combinations can be made from four different colored marbles if we use only three marbles at a time. The marbles are colored red, green, white, and yellow. 42. Two girls and their dates go to the drive-in, and each wants a different flavored ice cream cone. The drive-in has 24 flavors of ice cream. How many combinations of flavors may be chosen among the four of them if each one selects one flavor? 43. We want to paint three rooms in a house, each a different color, and we may choose from seven different colors of paint. How many color combinations are possible for the three rooms? 44. Find the number of ways to take 10 people and place them in 3 groups of 3 where order does not matter. 45. Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter. 46. Find the number of ways to arrange 5 objects that are chosen from a set of 7 different objects given that order matters. 47. What is the total number of possible 5-letter arrangements of the letters WHITE if each letter is used exactly once? 48. How many different 3-digit numerals can be made from the digits {4, 5, 6, 7, 8} if a digit can appear just once in a numeral? 49. We wish to know how many color combinations can be made from four different colored marbles if we use only three marbles at a time. The marbles are colored red, green, white, and yellow. 50. At the pizza place, there are 8 toppings that you can put on your pizza. If you can order any number of those 8 toppings, then how many different toppings could you possibly order?
5 5 Multinomial Counting Problems Formulas and Notation ( ) n Multinomial Coefficients: k 1,k 2,...,k m = n! k 1!k 2! k m!, given unordered groups of sizes k 1, k 2,..., k n such that k i = n, the multinomial coefficient gives the number ways to arrange n objects into these groups. If this is confusing, a string of binomial coefficients can be used instead, choosing each group one at a time rather than choosing all the groups at once. Problems 1. Ten consultants will be divided into three teams of three and leader. How many different possibilities for the team configurations exist? 2. A teacher divides a class of 20 students into five teams of 4. How many different possibilities exist? 3. A teacher divides a class of 20 students into four teams of 5. How many different possibilities exist? Compare this to your answer to the previous question. Are there more or fewer possibilities than before? 4. A dormitory hallway of 25 girls is selecting several groups to host socials. They need two groups of 4 girls, two groups of 3 girls, two groups of 2 girls and a group of 7 girls for the semester finale. How many possibilities exist? 5. Compare the expressions. Calculate the total number of unique orderings of the letters DREADED. Then calculate the number of ways seven boys can be divided up into a team of 3, a team of 2 and a pair of boys to not be on either team.
Introduction. Firstly however we must look at the Fundamental Principle of Counting (sometimes referred to as the multiplication rule) which states:
Worksheet 4.11 Counting Section 1 Introduction When looking at situations involving counting it is often not practical to count things individually. Instead techniques have been developed to help us count
More information6. In how many different ways can you answer 10 multiple-choice questions if each question has five choices?
Pre-Calculus Section 4.1 Multiplication, Addition, and Complement 1. Evaluate each of the following: a. 5! b. 6! c. 7! d. 0! 2. Evaluate each of the following: a. 10! b. 20! 9! 18! 3. In how many different
More informationSection The Multiplication Principle and Permutations
Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
More informationAlgebra II- Chapter 12- Test Review
Sections: Counting Principle Permutations Combinations Probability Name Choose the letter of the term that best matches each statement or phrase. 1. An illustration used to show the total number of A.
More informationChapter 2 Math
Chapter 2 Math 3201 1 Chapter 2: Counting Methods: Solving problems that involve the Fundamental Counting Principle Understanding and simplifying expressions involving factorial notation Solving problems
More informationSimple Counting Problems
Appendix F Counting Principles F1 Appendix F Counting Principles What You Should Learn 1 Count the number of ways an event can occur. 2 Determine the number of ways two or three events can occur using
More informationBayes stuff Red Cross and Blood Example
Bayes stuff Red Cross and Blood Example 42% of the workers at Motor Works are female, while 67% of the workers at City Bank are female. If one of these companies is selected at random (assume a 50-50 chance
More informationMAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
More information6.1.1 The multiplication rule
6.1.1 The multiplication rule 1. There are 3 routes joining village A and village B and 4 routes joining village B and village C. Find the number of different ways of traveling from village A to village
More informationUnit Nine Precalculus Practice Test Probability & Statistics. Name: Period: Date: NON-CALCULATOR SECTION
Name: Period: Date: NON-CALCULATOR SECTION Vocabulary: Define each word and give an example. 1. discrete mathematics 2. dependent outcomes 3. series Short Answer: 4. Describe when to use a combination.
More informationFundamental Counting Principle 2.1 Page 66 [And = *, Or = +]
Math 3201 Assignment 1 of 1 Unit 2 Counting Methods Name: Fundamental Counting Principle 2.1 Page 66 [And = *, Or = +] Identify the choice that best completes the statement or answers the question. 1.
More information6) A) both; happy B) neither; not happy C) one; happy D) one; not happy
MATH 00 -- PRACTICE TEST 2 Millersville University, Spring 202 Ron Umble, Instr. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all natural
More informationa) 2, 4, 8, 14, 22, b) 1, 5, 6, 10, 11, c) 3, 9, 21, 39, 63, d) 3, 0, 6, 15, 27, e) 3, 8, 13, 18, 23,
Pre-alculus Midterm Exam Review Name:. Which of the following is an arithmetic sequence?,, 8,,, b),, 6, 0,, c), 9,, 9, 6, d), 0, 6,, 7, e), 8,, 8,,. What is a rule for the nth term of the arithmetic sequence
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More informationTree Diagrams and the Fundamental Counting Principle
Objective: In this lesson, you will use permutations and combinations to compute probabilities of compound events and to solve problems. Read this knowledge article and answer the following: Tree Diagrams
More informationCreated by T. Madas COMBINATORICS. Created by T. Madas
COMBINATORICS COMBINATIONS Question 1 (**) The Oakwood Jogging Club consists of 7 men and 6 women who go for a 5 mile run every Thursday. It is decided that a team of 8 runners would be picked at random
More informationFundamental Counting Principle 2.1 Page 66 [And = *, Or = +]
Math 3201 Assignment 2 Unit 2 Counting Methods Name: Fundamental Counting Principle 2.1 Page 66 [And = *, Or = +] Identify the choice that best completes the statement or answers the question. Show all
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationFinite Mathematics MAT 141: Chapter 8 Notes
Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication
More informationSec. 4.2: Introducing Permutations and Factorial notation
Sec. 4.2: Introducing Permutations and Factorial notation Permutations: The # of ways distinguishable objects can be arranged, where the order of the objects is important! **An arrangement of objects in
More informationDiscrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
More informationFundamental Counting Principle
Lesson 88 Probability with Combinatorics HL2 Math - Santowski Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more
More information10.2.notebook. February 24, A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit.
Section 10.2 It is not always important to count all of the different orders that a group of objects can be arranged. A combination is a selection of r objects from a group of n objects where the order
More informationAdvanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY
Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY 1. Jack and Jill do not like washing dishes. They decide to use a random method to select whose turn it is. They put some red and blue
More informationCounting Principles Review
Counting Principles Review 1. A magazine poll sampling 100 people gives that following results: 17 read magazine A 18 read magazine B 14 read magazine C 8 read magazines A and B 7 read magazines A and
More informationPrinciples of Mathematics 12: Explained!
www.math12.com 284 Lesson 2, Part One: Basic Combinations Basic combinations: In the previous lesson, when using the fundamental counting principal or permutations, the order of items to be arranged mattered.
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.-9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
More information19.2 Permutations and Probability
Name Class Date 19.2 Permutations and Probability Essential Question: When are permutations useful in calculating probability? Resource Locker Explore Finding the Number of Permutations A permutation is
More informationFinite Math B, Chapter 8 Test Review Name
Finite Math B, Chapter 8 Test Review Name Evaluate the factorial. 1) 6! A) 720 B) 120 C) 360 D) 1440 Evaluate the permutation. 2) P( 10, 5) A) 10 B) 30,240 C) 1 D) 720 3) P( 12, 8) A) 19,958,400 B) C)
More informationMultiple Choice Questions for Review
Review Questions Multiple Choice Questions for Review 1. Suppose there are 12 students, among whom are three students, M, B, C (a Math Major, a Biology Major, a Computer Science Major. We want to send
More informationMathematics. Programming
Mathematics for the Digital Age and Programming in Python >>> Second Edition: with Python 3 Maria Litvin Phillips Academy, Andover, Massachusetts Gary Litvin Skylight Software, Inc. Skylight Publishing
More informationExercises Exercises. 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}?
Exercises Exercises 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}? 3. How many permutations of {a, b, c, d, e, f, g} end with
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationPERMUTATIONS AND COMBINATIONS
PERMUTATIONS AND COMBINATIONS 1. Fundamental Counting Principle Assignment: Workbook: pg. 375 378 #1-14 2. Permutations and Factorial Notation Assignment: Workbook pg. 382-384 #1-13, pg. 526 of text #22
More informationCSE 312: Foundations of Computing II Quiz Section #1: Counting
CSE 312: Foundations of Computing II Quiz Section #1: Counting Review: Main Theorems and Concepts 1. Product Rule: Suppose there are m 1 possible outcomes for event A 1, then m 2 possible outcomes for
More information2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)
2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are
More informationExam 2 Review (Sections Covered: 3.1, 3.3, , 7.1) 1. Write a system of linear inequalities that describes the shaded region.
Exam 2 Review (Sections Covered: 3.1, 3.3, 6.1-6.4, 7.1) 1. Write a system of linear inequalities that describes the shaded region. 5x + 2y 30 x + 2y 12 x 0 y 0 2. Write a system of linear inequalities
More informationNovember 8, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol
More informationWeek in Review #5 ( , 3.1)
Math 166 Week-in-Review - S. Nite 10/6/2012 Page 1 of 5 Week in Review #5 (2.3-2.4, 3.1) n( E) In general, the probability of an event is P ( E) =. n( S) Distinguishable Permutations Given a set of n objects
More informationPermutations & Combinations
Permutations & Combinations Extension 1 Mathematics HSC Revision UOW PERMUTATIONS AND COMBINATIONS: REVIEW 1. A combination lock has 4 dials each with 10 digits. How many possible arrangements are there?
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationHonors Precalculus Chapter 9 Summary Basic Combinatorics
Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each
More informationMath Week in Review #4
Math 166 Fall 2008 c Heather Ramsey and Joe Kahlig Page 1 Section 2.1 - Multiplication Principle and Permutations Math 166 - Week in Review #4 If you wish to accomplish a big goal that requires intermediate
More informationDetermine whether the given events are disjoint. 4) Being over 30 and being in college 4) A) No B) Yes
Math 34 Test #4 Review Fall 06 Name Tell whether the statement is true or false. ) 3 {x x is an even counting number} ) A) True False Decide whether the statement is true or false. ) {5, 0, 5, 0} {5, 5}
More informationThe Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n
Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product
More informationCS1802 Week 3: Counting Next Week : QUIZ 1 (30 min)
CS1802 Discrete Structures Recitation Fall 2018 September 25-26, 2018 CS1802 Week 3: Counting Next Week : QUIZ 1 (30 min) Permutations and Combinations i. Evaluate the following expressions. 1. P(10, 4)
More informationCSCI 2200 Foundations of Computer Science (FoCS) Solutions for Homework 7
CSCI 00 Foundations of Computer Science (FoCS) Solutions for Homework 7 Homework Problems. [0 POINTS] Problem.4(e)-(f) [or F7 Problem.7(e)-(f)]: In each case, count. (e) The number of orders in which a
More informationCOMBINATORIAL PROBABILITY
COMBINATORIAL PROBABILITY Question 1 (**+) The Oakwood Jogging Club consists of 7 men and 6 women who go for a 5 mile run every Thursday. It is decided that a team of 8 runners would be picked at random
More informationProblem Set 2. Counting
Problem Set 2. Counting 1. (Blitzstein: 1, Q3 Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his favorite ten restaurants. i
More informationaabb abab abba baab baba bbaa permutations of these. But, there is a lot of duplicity in this list, each distinct word (such as 6! 3!2!1!
Introduction to COMBINATORICS In how many ways (permutations) can we arrange n distinct objects in a row?answer: n (n ) (n )... def. = n! EXAMPLE (permuting objects): What is the number of different permutations
More informationATHS FC Math Department Al Ain Remedial worksheet. Lesson 10.4 (Ellipses)
ATHS FC Math Department Al Ain Remedial worksheet Section Name ID Date Lesson Marks Lesson 10.4 (Ellipses) 10.4, 10.5, 0.4, 0.5 and 0.6 Intervention Plan Page 1 of 19 Gr 12 core c 2 = a 2 b 2 Question
More informationPermutations and Combinations Practice Test
Name: Class: Date: Permutations and Combinations Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Suppose that license plates in the fictional
More informationACTIVITY 6.7 Selecting and Rearranging Things
ACTIVITY 6.7 SELECTING AND REARRANGING THINGS 757 OBJECTIVES ACTIVITY 6.7 Selecting and Rearranging Things 1. Determine the number of permutations. 2. Determine the number of combinations. 3. Recognize
More informationMath 3012 Applied Combinatorics Lecture 2
August 20, 2015 Math 3012 Applied Combinatorics Lecture 2 William T. Trotter trotter@math.gatech.edu The Road Ahead Alert The next two to three lectures will be an integrated approach to material from
More informationWEEK 7 REVIEW. Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.1)
WEEK 7 REVIEW Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.) Definition of Probability (7.2) WEEK 8-7.3, 7.4 and Test Review THE MULTIPLICATION
More informationUnit on Permutations and Combinations (Counting Techniques)
Page 1 of 15 (Edit by Y.M. LIU) Page 2 of 15 (Edit by Y.M. LIU) Unit on Permutations and Combinations (Counting Techniques) e.g. How many different license plates can be made that consist of three digits
More informationCPCS 222 Discrete Structures I Counting
King ABDUL AZIZ University Faculty Of Computing and Information Technology CPCS 222 Discrete Structures I Counting Dr. Eng. Farag Elnagahy farahelnagahy@hotmail.com Office Phone: 67967 The Basics of counting
More informationName: Exam 1. September 14, 2017
Department of Mathematics University of Notre Dame Math 10120 Finite Math Fall 2017 Name: Instructors: Basit & Migliore Exam 1 September 14, 2017 This exam is in two parts on 9 pages and contains 14 problems
More informationSolving Counting Problems
4.7 Solving Counting Problems OAL Solve counting problems that involve permutations and combinations. INVESIAE the Math A band has recorded 3 hit singles over its career. One of the hits went platinum.
More informationSection Summary. Permutations Combinations Combinatorial Proofs
Section 6.3 Section Summary Permutations Combinations Combinatorial Proofs Permutations Definition: A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement
More informationCHAPTER 8 Additional Probability Topics
CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information
More informationName: Exam I. February 5, 2015
Department of Mathematics University of Notre Dame Math 10120 Finite Math Spring 201 Name: Instructors: Garbett & Migliore Exam I February, 201 This exam is in two parts on 10 pages and contains 1 problems
More informationName: Spring P. Walston/A. Moore. Topic worksheet # assigned #completed Teacher s Signature Tree Diagrams FCP
Name: Spring 2016 P. Walston/A. Moore Topic worksheet # assigned #completed Teacher s Signature Tree Diagrams 1-0 13 FCP 1-1 16 Combinations/ Permutations Factorials 1-2 22 1-3 20 Intro to Probability
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More informationMATH 166 Exam II Sample Questions Use the histogram below to answer Questions 1-2: (NOTE: All heights are multiples of.05) 1. What is P (X 1)?
MATH 166 Exam II Sample Questions Use the histogram below to answer Questions 1-2: (NOTE: All heights are multiples of.05) 1. What is P (X 1)? (a) 0.00525 (b) 0.0525 (c) 0.4 (d) 0.5 (e) 0.6 2. What is
More informationCh. 12 Permutations, Combinations, Probability
Alg 3(11) 1 Counting the possibilities Permutations, Combinations, Probability 1. The international club is planning a trip to Australia and wants to visit Sydney, Melbourne, Brisbane and Alice Springs.
More informationSec 5.1 The Basics of Counting
1 Sec 5.1 The Basics of Counting Combinatorics, the study of arrangements of objects, is an important part of discrete mathematics. In this chapter, we will learn basic techniques of counting which has
More informationCHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY
CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many real-world fields, such as insurance, medical research, law enforcement, and political science. Objectives:
More informationCSE 312: Foundations of Computing II Quiz Section #1: Counting (solutions)
CSE 31: Foundations of Computing II Quiz Section #1: Counting (solutions Review: Main Theorems and Concepts 1. Product Rule: Suppose there are m 1 possible outcomes for event A 1, then m possible outcomes
More informationCounting (Enumerative Combinatorics) X. Zhang, Fordham Univ.
Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 Chance of winning?! What s the chances of winning New York Megamillion Jackpot!! just pick 5 numbers from 1 to 56, plus a mega ball number
More informationCHAPTER - 7 PERMUTATIONS AND COMBINATIONS KEY POINTS When a job (task) is performed in different ways then each way is called the permutation. Fundamental Principle of Counting : If a job can be performed
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1324 Test 3 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Insert " " or " " in the blank to make the statement true. 1) {18, 27, 32}
More informationLEVEL I. 3. In how many ways 4 identical white balls and 6 identical black balls be arranged in a row so that no two white balls are together?
LEVEL I 1. Three numbers are chosen from 1,, 3..., n. In how many ways can the numbers be chosen such that either maximum of these numbers is s or minimum of these numbers is r (r < s)?. Six candidates
More informationMath 1116 Probability Lecture Monday Wednesday 10:10 11:30
Math 1116 Probability Lecture Monday Wednesday 10:10 11:30 Course Web Page http://www.math.ohio state.edu/~maharry/ Chapter 15 Chances, Probabilities and Odds Objectives To describe an appropriate sample
More informationAlgebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations
Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)
More informationMat 344F challenge set #2 Solutions
Mat 344F challenge set #2 Solutions. Put two balls into box, one ball into box 2 and three balls into box 3. The remaining 4 balls can now be distributed in any way among the three remaining boxes. This
More informationMATH 351 Fall 2009 Homework 1 Due: Wednesday, September 30
MATH 51 Fall 2009 Homework 1 Due: Wednesday, September 0 Problem 1. How many different letter arrangements can be made from the letters BOOKKEEPER. This is analogous to one of the problems presented in
More informationLesson1.notebook July 07, 2013
Lesson1.notebook July 07, 2013 Topic: Counting Principles Today's Learning goal: I can use tree diagrams, Fundamental counting principle and indirect methods to determine the number of outcomes. Tree Diagram
More informationSolutions to Exercises on Page 86
Solutions to Exercises on Page 86 #. A number is a multiple of, 4, 5 and 6 if and only if it is a multiple of the greatest common multiple of, 4, 5 and 6. The greatest common multiple of, 4, 5 and 6 is
More informationProbability Study Guide Date Block
Probability Study Guide Name Date Block In a regular deck of 52 cards, face cards are Kings, Queens, and Jacks. Find the following probabilities, if one card is drawn: 1)P(not King) 2) P(black and King)
More informationCounting and Probability Math 2320
Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More informationFinite Math Section 6_4 Solutions and Hints
Finite Math Section 6_4 Solutions and Hints by Brent M. Dingle for the book: Finite Mathematics, 7 th Edition by S. T. Tan. DO NOT PRINT THIS OUT AND TURN IT IN!!!!!!!! This is designed to assist you in
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 1 6
Math 300 Exam 4 Review (Chapter 11) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Give the probability that the spinner shown would land on
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationSolutions for Exam I, Math 10120, Fall 2016
Solutions for Exam I, Math 10120, Fall 2016 1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 3} B = {2, 4, 6, 8, 10}. C = {4, 5, 6, 7, 8}. Which of the following sets is equal to (A B) C? {1, 2, 3,
More informationMAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions
MAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions 1. Appetizers: Salads: Entrées: Desserts: 2. Letters: (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U,
More informationPermutations and Combinations. Quantitative Aptitude & Business Statistics
Permutations and Combinations Statistics The Fundamental Principle of If there are Multiplication n 1 ways of doing one operation, n 2 ways of doing a second operation, n 3 ways of doing a third operation,
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Statistics Homework Ch 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
More informationSTAT 430/510 Probability Lecture 1: Counting-1
STAT 430/510 Probability Lecture 1: Counting-1 Pengyuan (Penelope) Wang May 22, 2011 Introduction In the early days, probability was associated with games of chance, such as gambling. Probability is describing
More information11.3B Warmup. 1. Expand: 2x. 2. Express the expansion of 2x. using combinations. 3. Simplify: a 2b a 2b
11.3 Warmup 1. Expand: 2x y 4 2. Express the expansion of 2x y 4 using combinations. 3 3 3. Simplify: a 2b a 2b 4. How many terms are there in the expansion of 2x y 15? 5. What would the 10 th term in
More information* Order Matters For Permutations * Section 4.6 Permutations MDM4U Jensen. Part 1: Factorial Investigation
Section 4.6 Permutations MDM4U Jensen Part 1: Factorial Investigation You are trying to put three children, represented by A, B, and C, in a line for a game. How many different orders are possible? a)
More informationMath 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
More informationMAT3707. Tutorial letter 202/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/202/1/2017
MAT3707/0//07 Tutorial letter 0//07 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Semester Department of Mathematical Sciences SOLUTIONS TO ASSIGNMENT 0 BARCODE Define tomorrow university of south africa
More informationUnit 5 Radical Functions & Combinatorics
1 Graph of y Unit 5 Radical Functions & Combinatorics x: Characteristics: Ex) Use your knowledge of the graph of y x and transformations to sketch the graph of each of the following. a) y x 5 3 b) f (
More informationPERMUTATIONS AND COMBINATIONS
8 PERMUTATIONS AND COMBINATIONS FUNDAMENTAL PRINCIPLE OF COUNTING Multiplication Principle : If an operation can be performed in 'm' different ways; following which a second operation can be performed
More informationPermutations and Combinations Section
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Permutations and Combinations Section 13.3-13.4 Dr. John Ehrke Department of Mathematics Fall 2012 Permutations A permutation
More informationSTATISTICAL COUNTING TECHNIQUES
STATISTICAL COUNTING TECHNIQUES I. Counting Principle The counting principle states that if there are n 1 ways of performing the first experiment, n 2 ways of performing the second experiment, n 3 ways
More information