# Introduction. Firstly however we must look at the Fundamental Principle of Counting (sometimes referred to as the multiplication rule) which states:

Size: px
Start display at page:

Download "Introduction. Firstly however we must look at the Fundamental Principle of Counting (sometimes referred to as the multiplication rule) which states:"

Transcription

1 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 efficiently and accurately. In particular this chapter looks at permutations and combinations. Firstly however we must look at the Fundamental Principle of Counting (sometimes referred to as the multiplication rule) which states: If there are m ways of doing one thing and n ways of doing another, then there are m n ways of doing the first thing followed by the second. This rule can best be understood by looking at an example. Example 1 : There are 3 t-shirts and 2 pairs of jeans in the cupboard. How many possible outfits are there? For each t-shirt there are 2 possibilities of jeans. All together we have three lots of these two possibilities, or, 3 2 = 6. Example 2 : An ice-cream shop offers 3 types of cones and 5 different flavours of ice-cream. How many possible ice-cream cone combinations are there? For each of the 3 cones there are 5 possible toppings so altogether there are 3 5 = 15 possible ice-cream cone combinations. Exercises: 1. The local pizzeria offers a choice of 2 pizzas - supreme or vegetarian, 3 sides - chips, salad or coleslaw, and 4 drinks - juice, coke, ginger beer or water. For dinner I decide to have 1 pizza, 1 side, and 1 drink. How many possible meals do I have to choose from? 2. How many different car number plates can be made if each is to display 3 letters followed by 3 numbers? 3. Your friend wants to perform a magic trick and asks you to draw 2 cars from a standard deck of 52. The first card you draw must be placed face down and the second placed face up on the table. How many ways are there of drawing the 2 cards? 1

2 Section 2 Counting Techniques For the many circumstances where we need to count the number of outcomes there are two different counting situations - permutations and combinations. A permutation is an arrangement where the order of selection matters. A combination is an arrangement where the order of selection doesn t matter. Example 1 : The number of ways of arranging 5 people in a line. There are 5 choices for the first spot, 4 choices for the second and so on, so we have = 5!. Therefore the number of ways of arranging n things into n places is n!. Example 2 : To arrange 3 people out of 5 in a line there are = 5! ways. 2! These are all permutations, the number of ways of choosing k things out of n where the order matters is n! = (n k)! = n P k. How about choosing 3 out of 5 people to put on a committee? This time, it doesn t matter whether you are chosen first, second or third, you are still on the committee. There are ways of choosing 3 people in order. However since order doesn t matter we have over counted, and therefore we need to divide by the number of ways of arranging the three people that are chosen, 3!. So the number of ways of choosing this committee is the number of ways of choosing with order, divided by = 5! 2!3!. This is a combination, the number of ways of choosing k things from n where order does NOT matter n! = k!(n k)! = n C k. Definition 1 : A permutation is the number of ways of choosing k things out of a possible n, where the order that they are chosen matters and is notated n P k. n P k = n(n 1)(n 2)... (n k + 1) = n! (n k)!. 2

3 Definition 2 : A combination is the number of ways of choosing k things out of a possible n, where the order that they are chosen does not matter and is notated n C k. n C k = n(n 1)(n 2)... (n k + 1) k(k 1)... 1 = n! (n k)!k!. These definitions are best illustrated by examples which we ll cover below and in the next 2 sections. Example 3 : (a) A teacher wants to randomly choose 5 people from the class of 30 to help out at the open day BBQ. In how many ways can this be done? (b) A teacher wants to award prizes for 1st, 2nd, 3rd, 4th and 5th in the class of 30. In how many ways can the prizes be awarded (assume no two students tie)? Answers: (a) In the first case, the 5 people are chosen, and it doesn t matter whether a person is chosen first, second or fifth, they all receive the same extra work, and so there are 30 C 5 = ways to choose the students with those conditions. = 142, 506 (b) Here the order that the student are chosen does matter, and so there are 30 P 5 = = 17, 100, 720 ways to choose the students with those conditions. This is a significantly more as expected. Notice the short way of calculating 30 P 5 and 30 C 5 applies to all such calculations. To find 30 P 5 you can just multiply 5 consecutive descending numbers together, starting at 30. Similarly calculating 30 C 5 can be done by multiplying 5 consecutive descending numbers together, starting at 30, and then dividing by the product of 1 through 5. 3

4 Exercises: 1. Calculate 10 P 4 and 10 C Calculate 7 P 5 and 7 C In an 18 team league, how many ways can the 8 teams for the finals be decided? In how many ways can the first 4 positions be decided? 4. A child wants to draw a picture using only three different colours from a set containing twelve different colour pencils. In how many ways can the colours be chosen? 5. In the front of a building there are three doors each to be painted a different colour from twelve different available colours. How many colour arrangements for the doors are there? Section 3 Permutations Example 1 : (a) How many four digit numbers can be formed using only the digits 1, 2, 3, 4, 5, 6? (b) How many four digit numbers from (a) have no repeated digits? (c) How many four digit numbers from (b) are greater than 5000? Answers: (a) There are 6 possible digits for each of the four places in the number, so there are = 6 4 = 1296 of these numbers. (b) There are 6 digits for the first place, and then only 5 digits for the second and so on. So there are = 360 such numbers. (c) There are 2 choices for the first digit (a 5 or a 6), then 5 choices, 4 choices and 3 choices respectively for the remaining digits. So there are = 120 such numbers. Example 2 : Three adults and five children are seated randomly in a row. (a) In how many ways can this be done? (b) In how many ways can this be done if the three adults are seated together? (c) In how many ways can this be done if the three adults are seated together and the five children are also seated together. 4

5 Answers: (a) There are 8! ways of arranging 8 people in a row. (b) There are 3! ways of arranging the adults. We now need to arrange 6 objects (1 group of adults and 5 individual children) in a row. Therefore the answer is 3! 6! ways. (c) There are 3! ways of arranging the adults, 5! ways of arranging the children, and 2! ways of arranging the 2 groups. So the answer is 3! 5! 2!. Permutations with Repeated Objects Example 3 : How many arrangements of the letters of the word IRRIGATION are there? Answer: There are 10 letters. If the letters were all different there would be 10! arrangements. However there are three I s and two R s and so we need to divide by 3! 2!. Therefore the answer is 10! 3!2!. Example 4 : In how many ways can we rearrange the letters in MATHS IS FUN (a) with no restrictions? (b) if the first and last letter must be vowels? Answers: (a) There are 10 letters to rearrange. Two of them are S s and so that we do not over count we need to divide by the number of ways of arranging the S, so there are 10! ways to arrange these letters. 2! (b) In the second instance, we first select the vowels and there are 3 2 ways of the selecting the first and last as vowels. Then we have 8 letters left with 2 S s repeating so there are 8! 8! ways to do this. Hence there are 3 2 2! ways 2! of arranging these letters so that the first and last are vowels. Exercises: 1. How many 4 digit numbers can be formed from the digits 1,2,3,4 if (a) repetitions are allowed? 5

6 (b) repetitions are not allowed? 2. How many 3 digit arrangements can be formed from the digits 0,1,2,3,4,5,6,7 and 8? 3. Seven people are to occupy consecutive seats in a theatre. In how many ways can this be done if (a) there are no restrictions? (b) two people A and B sit at opposite ends of the row of seven seats? (c) two people A and B sit together? (d) two people A and B do not sit together? 4. How many arrangements of the letters in the following words are possible? (a) SYDNEY (b) GEOMETRY (c) EXCELLENCE Section 4 Combinations Example 1 : A student must select 6 subjects. In how many ways can they do that if there are 13 subjects and 1 is compulsory? Since one subject is compulsory the student must select 5 subjects from 12, there are C 5 = = 792, ways to do this. Example 2 : In a lottery you select 6 numbers out of 40, how many ways are there to do this? You are selecting 6 things from 40 with order not mattering, thus there are 40 C 6 = = 3, 838, 380ways. Example 3 : You want to choose a committee of 5 people from 7 men and 8 women. (a) How many ways can this be done? (b) How many ways can this be done if you want a majority of women on the committee? In (a) it doesn t matter if the 5 people are men or women. So we are choosing 5 people from 15 and there are 15 C 5 ways of doing this. In (b) we need either 3 women (and 2 men) or 4 women (and 1 man) or 5 women. So there are 8 C 3 7 C C 4 7 C C 5 ways of doing this. 6

7 Exercises: 1. In how many ways can you choose 2 chocolates from a bag containing 6 different chocolates? 2. Twelve dots are spaced equally around a circle. How many different triangles can be formed by joining dots? 3. Tickets are numbered from 1 to tickets are chosen. In how many ways can this be done if the selection contains (a) all odd numbers? (b) 3 odd numbers and 3 even numbers? (c) the numbers 1 and 2? 7

8 Exercises for Worksheet Decide whether or not order of selection is important and then calculate the following. (a) How many different sets of three colours can be selected from the colours red, orange, yellow, green, blue, and violet? (b) In how many ways can a team of five basketball players be selected from 8 girls? (c) A race has 8 runners. In how many ways can the first three places be decided? (d) A secretary has nine letters and only five stamps. How many ways can he select the letters for posting? 2. How many different possible full house (one pair, one three of a kind) hands are there in 5 card poker? 3. We choose 12 cards from the usual deck of 52 playing cards. (a) How many different ways can this be done? (b) How many ways can it be done if they must all come from the same suit? (c) How many ways can it be done if we need exactly 3 kings and 3 queens? (d) How many ways can it be done if all cards must have different face values? 4. There are 10 boys and 11 girls at a school. (a) How many different ways can the boys each choose a girl to take to the formal? (b) One of the girls doesn t want to go to the formal, how many ways are there to make this choice now? 5. A town of 30 people is to choose a committee of 3 to represent them, how many different ways can this be done? How many ways can it be done if one person is to be the chairperson, one the treasurer and one the secretary? 6. In a certain electorate there are 6 candidates: labor, liberal, greens, and three independents. Their names are to be placed in random order on the ballot paper. In how many ways can this be done if (a) the labor candidate comes first? (b) the liberal candidate comes first? (c) the three independent candidates are together? 7. From a class of 30 students, five students are to be selected to complete a survey. In how many ways can the choice be made? 8

9 8. A 4 digit password is to be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. How many different passwords are possible (a) if the digits are repeated? (b) if there are no repeated digits? 9. How many different arrangements of the word ELLIPSE are possible if (a) there are no restrictions? (b) the arrangement starts with S? (c) both L s are together? (d) the letters are in alphabetical order? 10. Seven people sit in a circle. How many ways can this be done if (a) there are no restrictions? (b) two people A and B sit together? (c) three people A, B and C sit together? 11. A committee of 5 is to be chosen from 4 men and 6 women. In how many ways can this be done if (a) there are no restrictions? (b) the committee consists of women only? (c) there is at least one man? (d) there is a majority of women? cards are to be chosen from a standard 52-card deck. In how many ways can this be done if (a) all of the cards are clubs? (b) all of the cards are of the same suit? (c) there are three clubs and two spades? (d) there are three of one suit and two of another? 13. A bag contains 5 red, 6 blue and 4 yellow marbles. Three are drawn out at random. In how many ways can they be drawn so that (a) are all blue? (b) are all the same colour? (c) are all different colours? 9

10 14. In order to be photographed 10 people stand in two rows of 5, one in front of the other. In how many ways can this be done if (a) there are no restrictions? (b) A and B are to be in the front row? (c) A and B are in different rows? 15. In how many ways can you choose 10 people out of 30 to sit on a bench in order, if two particular people must be selected and seated together? 16. You are to pick three singers for the Superbowl, one to sing before the game, one to sing at the end of the game, and one to sing at halftime. If there are 12 applicants, how many different possibilities are there if each singer can only sing once? 10

11 Answers for Worksheet 4.11 Section Section , , , C P 3 Section 3 1. (a) 256 (b) (a) 5040 (b) 240 (c) 1440 (d) (a) 360 (b) (c) Section C 2 3. (a) 13 C 6 (b) 13 C 3 12 C C 3 (c) 23 C 4 11

12 Exercises (a) Not important, 6 C 3 (b) Not important, 8 C 5 (c) Important, 8 P 3 (d) Important (if stamps are distinct), 9 P C 1 4 C 3 12 C 1 4 C 2 3. (a) 52 C 12 (b) 4 C 1 13 C 12 (c) 4 C 3 4 C 3 44 C 6 (d) 13 C (a) 11 P 10 (b) 10! C 3, 30 P 3 6. (a) 5! (b) 5! (c) 3! 4! C 5 8. (a) 10 4 (b) 10 P 4 9. (a) 7! 2!2! (b) 6! 2!2! (c) 6! 2! (d) 2! 2! 10. (a) 7! 7 (b) 2! 6! 6 (c) 3! 5! (a) 10 C 5 (b) 6 C 5 (c) 246 (d) (a) 13 C 5 (b) 4 C 1 13 C 5 (c) 13 C 3 13 C 2 (d) 4 C 1 13 C 3 3 C 1 13 C (a) 6 C 3 (b) 5 C C C 3 (c) (a) 10! (b) 5 P 2 8! (c) ! 15. 2! 28 C 8 9! C 3 3! 12

### Advanced 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

### 50 Counting Questions

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!,

### April 10, ex) Draw a tree diagram of this situation.

April 10, 2014 12-1 Fundamental Counting Principle & Multiplying Probabilities 1. Outcome - the result of a single trial. 2. Sample Space - the set of all possible outcomes 3. Independent Events - when

### Permutations & 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?

### Principles 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.

### 6.1.1 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

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

### Section 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

### COMBINATORIAL 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

### Fundamentals 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

### 6. 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

### Algebra 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.

### Principles of Counting

Name Date Principles of Counting Objective: To find the total possible number of arrangements (ways) an event may occur. a) Identify the number of parts (Area Codes, Zip Codes, License Plates, Password,

### Unit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)

Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,

### Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11

Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value

### Simple 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

### JUST THE MATHS UNIT NUMBER PROBABILITY 2 (Permutations and combinations) A.J.Hobson

JUST THE MATHS UNIT NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by A.J.Hobson 19.2.1 Introduction 19.2.2 Rules of permutations and combinations 19.2.3 Permutations of sets with some objects

### Solutions 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,

### Sec. 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

### CONTENTS CONTENTS PAGES 11.0 CONCEPT MAP A. PERMUTATIONS a EXERCISE A B. COMBINATIONS a EXERCISE B PAST YEAR SPM

PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM 5 MODULE 11 PERMUTATIONS AND COMBINATIONS 0 CONTENTS CONTENTS PAGES 11.0 CONCEPT MAP 2 11.1 A. PERMUTATIONS 3 11.1a EXERCISE A.1 3 11.2

### CHAPTER 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:

### Fundamental 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

### 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}?

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

### Independent Events. If we were to flip a coin, each time we flip that coin the chance of it landing on heads or tails will always remain the same.

Independent Events Independent events are events that you can do repeated trials and each trial doesn t have an effect on the outcome of the next trial. If we were to flip a coin, each time we flip that

### PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (d) (e)...

Math 10120, Exam I September 15, 2016 The Honor Code is in e ect for this examination. All work is to be your own. You may use a calculator. The exam lasts for 1 hour and 15 min. Be sure that your name

### LC OL Probability. ARNMaths.weebly.com. As part of Leaving Certificate Ordinary Level Math you should be able to complete the following.

A Ryan LC OL Probability ARNMaths.weebly.com Learning Outcomes As part of Leaving Certificate Ordinary Level Math you should be able to complete the following. Counting List outcomes of an experiment Apply

### Determine 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}

### MULTIPLE 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

### Counting Methods. Mathematics 3201

Mathematics 3201 Unit 2 2.1 - COUNTING PRINCIPLES Goal: Determine the Fundamental Counting Principle and use it to solve problems. Example 1: Hannah plays on her school soccer team. The soccer uniform

### Topic: Probability Problems Involving AND & OR- Worksheet 1

Topic: Probability Problems Involving AND & OR- Worksheet 1 1. In a game a die numbered 9 through 14 is rolled. What is the probability that the value of a roll will be a multiple of two or ten? 2. Mark

### Unit 5 Radical Functions & Combinatorics

1 Unit 5 Radical Functions & Combinatorics General Outcome: Develop algebraic and graphical reasoning through the study of relations. Develop algebraic and numeric reasoning that involves combinatorics.

### Counting 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

### Chapter 1 - Set Theory

Midterm review Math 3201 Name: Chapter 1 - Set Theory Part 1: Multiple Choice : 1) U = {hockey, basketball, golf, tennis, volleyball, soccer}. If B = {sports that use a ball}, which element would be in

### MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG

MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices

### Determine the number of permutations of n objects taken r at a time, where 0 # r # n. Holly Adams Bill Mathews Peter Prevc

4.3 Permutations When All Objects Are Distinguishable YOU WILL NEED calculator standard deck of playing cards EXPLORE How many three-letter permutations can you make with the letters in the word MATH?

CHAPTER - 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

### 1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,

### 2. How many even 4 digit numbers can be made using 0, 2, 3, 5, 6, 9 if no repeats are allowed?

Math 30-1 Combinatorics Practice Test 1. A meal combo consists of a choice of 5 beverages, main dishes, and side orders. The number of different meals that are available if you have one of each is A. 15

### Unit 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 (

### WEEK 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

### Math 166: Topics in Contemporary Mathematics II

Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define

### Question No: 1 If you join all the vertices of a heptagon, how many quadrilaterals will you get?

Volume: 427 Questions Question No: 1 If you join all the vertices of a heptagon, how many quadrilaterals will you get? A. 72 B. 36 C. 25 D. 35 E. 120 Question No: 2 Four students have to be chosen 2 girls

### Fundamental 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.

### Block 1 - Sets and Basic Combinatorics. Main Topics in Block 1:

Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.

### Algebra II Probability and Statistics

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Slide 3 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional Probability

### Counting 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

### Functional Skills Mathematics

Functional Skills Mathematics Level Learning Resource Probability D/L. Contents Independent Events D/L. Page - Combined Events D/L. Page - 9 West Nottinghamshire College D/L. Information Independent Events

### Permutations 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,

### Finite 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)

### Algebra II. Sets. Slide 1 / 241 Slide 2 / 241. Slide 4 / 241. Slide 3 / 241. Slide 6 / 241. Slide 5 / 241. Probability and Statistics

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Slide 3 / 241 Slide 4 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional

### PROBABILITY. Example 1 The probability of choosing a heart from a deck of cards is given by

Classical Definition of Probability PROBABILITY Probability is the measure of how likely an event is. An experiment is a situation involving chance or probability that leads to results called outcomes.

### Algebra II. Slide 1 / 241. Slide 2 / 241. Slide 3 / 241. Probability and Statistics. Table of Contents click on the topic to go to that section

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 241 Sets Independence and Conditional Probability

### Name: 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

### Exam 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

### LEVEL 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

### Elementary Combinatorics

184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are

### a) 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

### Bayes 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

### CISC 1400 Discrete Structures

CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest

### Lesson1.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

### PERMUTATIONS 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

### MGF 1106: Exam 2 Solutions

MGF 1106: Exam 2 Solutions 1. (15 points) A coin and a die are tossed together onto a table. a. What is the sample space for this experiment? For example, one possible outcome is heads on the coin and

### STAT 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

### Probability 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

### MATH 13150: Freshman Seminar Unit 4

MATH 1150: Freshman Seminar Unit 1. How to count the number of collections The main new problem in this section is we learn how to count the number of ways to pick k objects from a collection of n objects,

### Solving 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.

### 4.1 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

### Section 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

### Ch. 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.

### Unit 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

### Math 1 Unit 4 Mid-Unit Review Chances of Winning

Math 1 Unit 4 Mid-Unit Review Chances of Winning Name My child studied for the Unit 4 Mid-Unit Test. I am aware that tests are worth 40% of my child s grade. Parent Signature MM1D1 a. Apply the addition

### Course Learning Outcomes for Unit V

UNIT V STUDY GUIDE Counting Reading Assignment See information below. Key Terms 1. Combination 2. Fundamental counting principle 3. Listing 4. Permutation 5. Tree diagrams Course Learning Outcomes for

### MIND ACTION SERIES THE COUNTING PRINCIPLE AND PROBABILITY GRADE

MIND ACTION SERIES THE COUNTING PRINCIPLE AND PROBABILITY GRADE 12 MARK PHILLIPS THE COUNTING PRINCIPLE AND PROBABILITY GRADE 12 1. The basic product rule of choices: a1 a2 a3... an 2. The product rule

### Section 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

### NAME DATE PERIOD. Study Guide and Intervention

9-1 Section Title The probability of a simple event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. Outcomes occur at random if each outcome occurs by chance.

### Combinatorics problems

Combinatorics problems Sections 6.1-6.4 Math 245, Spring 2011 1 How to solve it There are four main strategies for solving counting problems that we will look at: Multiplication principle: A man s wardrobe

### Permutations 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

### Fundamental 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

### 4.1 Organized Counting McGraw-Hill Ryerson Mathematics of Data Management, pp

Name 4.1 Organized Counting McGraw-Hill yerson Mathematics of Data Management, pp. 225 231 1. Draw a tree diagram to illustrate the possible travel itineraries for Pietro if he can travel from home to

### Math 1101 Combinations Handout #17

Math 1101 Combinations Handout #17 1. Compute the following: (a) C(8, 4) (b) C(17, 3) (c) C(20, 5) 2. In the lottery game Megabucks, it used to be that a person chose 6 out of 36 numbers. The order of

### Unit 14 Probability. Target 3 Calculate the probability of independent and dependent events (compound) AND/THEN statements

Target 1 Calculate the probability of an event Unit 14 Probability Target 2 Calculate a sample space 14.2a Tree Diagrams, Factorials, and Permutations 14.2b Combinations Target 3 Calculate the probability

### W = {Carrie (U)nderwood, Kelly (C)larkson, Chris (D)aughtry, Fantasia (B)arrino, and Clay (A)iken}

UNIT V STUDY GUIDE Counting Course Learning Outcomes for Unit V Upon completion of this unit, students should be able to: 1. Apply mathematical principles used in real-world situations. 1.1 Draw tree diagrams

### Mathematics. 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

### 2.5 Sample Spaces Having Equally Likely Outcomes

Sample Spaces Having Equally Likely Outcomes 3 Sample Spaces Having Equally Likely Outcomes Recall that we had a simple example (fair dice) before on equally-likely sample spaces Since they will appear

### Chapter 10A. a) How many labels for Product A are required? Solution: ABC ACB BCA BAC CAB CBA. There are 6 different possible labels.

Chapter 10A The Addition rule: If there are n ways of performing operation A and m ways of performing operation B, then there are n + m ways of performing A or B. Note: In this case or means to add. Eg.

### ATHS 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

### In how many ways can a team of three snow sculptors be chosen to represent Amir s school from the nine students who have volunteered?

4.6 Combinations GOAL Solve problems involving combinations. LEARN ABOUT the Math Each year during the Festival du Voyageur, held during February in Winnipeg, Manitoba, high schools compete in the Voyageur

### Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1

Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..

### { a, b }, { a, c }, { b, c }

12 d.) 0(5.5) c.) 0(5,0) h.) 0(7,1) a.) 0(6,3) 3.) Simplify the following combinations. PROBLEMS: C(n,k)= the number of combinations of n distinct objects taken k at a time is COMBINATION RULE It can easily

### Mat 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

### Name: Section: Date:

WORKSHEET 5: PROBABILITY Name: Section: Date: Answer the following problems and show computations on the blank spaces provided. 1. In a class there are 14 boys and 16 girls. What is the probability of

### OCR Statistics 1. Probability. Section 2: Permutations and combinations. Factorials

OCR Statistics Probability Section 2: Permutations and combinations Notes and Examples These notes contain subsections on Factorials Permutations Combinations Factorials An important aspect of life is

### ACTIVITY 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

### Unit 5, Activity 1, The Counting Principle

Unit 5, Activity 1, The Counting Principle Directions: With a partner find the answer to the following problems. 1. A person buys 3 different shirts (Green, Blue, and Red) and two different pants (Khaki

### Probability, Permutations, & Combinations LESSON 11.1

Probability, Permutations, & Combinations LESSON 11.1 Objective Define probability Use the counting principle Know the difference between combination and permutation Find probability Probability PROBABILITY:

### Permutations and Combinations

Permutations and Combinations In statistics, there are two ways to count or group items. For both permutations and combinations, there are certain requirements that must be met: there can be no repetitions

### A magician showed a magic trick where he picked one card from a standard deck. Determine What is the probability that the card will be a queen card?

Topic : Probability Word Problems- Worksheet 1 What is the probability? 1. 2. 3. 4. Jill is playing cards with her friend when she draws a card from a pack of 20 cards numbered from 1 to 20. What is the

### Math 3201 Notes Chapter 2: Counting Methods

Learning oals: See p. 63 text. Math 30 Notes Chapter : Counting Methods. Counting Principles ( classes) Outcomes:. Define the sample space. P. 66. Find the sample space by drawing a graphic organizer such