Applied Math 40S Unit 6 Counting Methods

Lesson One: Counting Principles Lesson Two: Factorial Notation and Permutations Lesson Three: Permutations Involving Identical Objects Lesson Four: Permutations Involving Restrictions Lesson Five: Permutations Involving Groups Lesson Six: Combinations Lesson Seven: Distinguishing between Permutation and

Combination Problems Lesson Eight: Review Assignment Printed on: June 2019 Winnipeg Adult Education Center

Counting Methods Overview

These are the outcomes we will be learning in this unit. Check off each box once you feel confident with each outcome: At the end of this Unit, I can:

Represent and solve counting problems using a graphic organizer.

Understand the fundamental counting principle (FCP).

Solve counting problems.

Use the FCP to determine the number of permutations possible.

Determine the number of permutations possible when some elements are identical.

Determine the number of permutations when there are restrictions.

Determine the number of permutations when some objects are grouped.

Determine the complement of a permutation problem.

Determine the number of combinations possible.

Distinguish between permutations and combinations.

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Lesson 1: Counting Principles

We are constantly being asked to make choices or decisions. We must make choices about courses we want to take, what we want to order for a meal, or what we want our new PIN number or password to be. When two or more decisions must be made together, it helps to have a logical system that will allow us to consider all of these possibilities. In these situations, it helps to use a graphic organizer such as a chart, a tree diagram, or a Venn diagram.

Example 1 A school lunch program offers a soup and sandwich combo. Students must choose from one of two choices of soup (tomato, vegetable) and three choices of sandwich (egg, ham, cheese). Using a graphic organizer, list all possible lunch combos.

Example 2 You are completing a survey with three questions. The possible answers to each question are yes or no.

List all the possible outcomes of your responses to the survey.

GOAL:

• Represent and solve counting problems using a graphic organizer.

• Generalize, from examples, the fundamental counting principle.

• Solve counting problems using the fundamental counting principle.

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You can see that showing all possible outcomes for a task involving several decisions with a several choices for each decision can become unmanageable. In many situations, we are not interested in listing the actual specific outcomes but we are just interested in how many different outcomes are possible. Fortunately, there is a more efficient way to determine the number of different arrangements of a certain number of objects rather than individually listing and then counting them. This more efficient way is called the Fundamental Counting Principle.

The Fundamental Counting Principle states that:

If one task can be performed in “𝑚” ways and another task can be performed in “𝑛” ways, then the two tasks together can be performed in 𝑚 × 𝑛 ways.

Example 4 You are packing a wardrobe for an upcoming weekend trip. You pack two different colours of pants

(black and blue), three different shirts (red, white, and grey) and two different jackets (green and white).

Determine the number of different wardrobe combinations possible by wearing one pair of pants, one

shirt and one jacket.

Example 5 You are driving from your small town to the Big City and then you will continue on to visit your grandparents’ village. There are two roads to choose from between your town and the Big City and five roads to choose from between the Big City and your grandparents’ village. How many different ways are there to complete your journey?

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Assignment 1 1. Julie just had a new baby. She received three shirts (green, yellow, orange) and two pairs of leggings

(blue and white) for the baby. a) Use the Fundamental Counting Principle to determine the number of outfits for the baby if an

outfit consists of one shirt and one pair of leggings. b) To verify your count in part a), use a graphic organizer to list all the possible outfits.

2. Your sister is buying her first car. She has to choose between four different colours (red, black, white, silver) and two different interiors (leather or cloth). a) Use the Fundamental Counting Principle to determine the number of possible outcomes. b) To verify your count in part a), construct a graphic organizer to list all the possible outcomes.

3. A sportswear company makes ski pants in five different colours and four different sizes. How many different colour-size variations of ski pants does this company make?

4. A local diner offers a lunch special with a choice of 3 soups, 5 sandwiches, 4 drinks, and 2 desserts. How many meal variations are possible if you must choose one item from each category?

5. A pizza shop offers the following choices:

Charles would like to create a pizza by choosing one item from each category. Determine how many different pizzas can be made.

6. A survey consists of 5 questions. Each question can be answered true or false. How many different

ways are there for a student to complete the survey?

7. A multiple choice test has 4 questions with 3 possible answers for each question. How many different ways are there for a student to complete the test?

8. There are two roads from Winnipeg to Brandon and four roads from Brandon to Saskatoon. a) How many different routes could you take driving from Winnipeg to Saskatoon if you must travel

through Brandon? b) How many different routes could you take driving from Winnipeg to Saskatoon through Brandon

and back home again if you can’t drive through the same road twice?

9. There are 4 children waiting to have their photo taken. a) How many choices are there for who goes first? b) Of the remaining children, how many choices are there for who goes next? c) If the children formed a line consisting of 4 positions, how many arrangements are possible?

10. In a school there is one front door, two side doors, and two back doors. What is the probability that a randomly selected student will enter through the front door and then exit out of either one of the back doors?

• 3 types of crust • 2 types of sauce

• 5 types of cheese • 6 meat toppings

• 8 vegetable toppings

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Lesson 2: Factorial Notation and Permutations

An arrangement of a set of objects is called a permutation. In a permutation, the order of the arrangement matters, like a combination to a lock or a pin number on a bank card.

Example 1 In how many ways can the letters in the word AMIGOS be re-arranged (or ‘permuted’)? Use the fundamental counting principle to determine your answer. Fortunately, there is a more efficient way to enter this multiplication into a calculator. It’s called Factorial Notation. In mathematics, the factorial of a non-negative integer 𝑛, denoted by n! is the product of all positive integers less than or equal to 𝑛. Using factorial notation, the product 6 × 5 × 4 × 3 × 2 × 1 can be represented by 6! (read as “six factorial”). Use the factorial key on your calculator to verify that 6! = 720.

Note: on the graphing calculator, the factorial symbol can be accessed by pressing the MATH button and then using the cursor to select PRB; the factorial symbol is option 4.

In general, for any whole number 𝑛:

𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) × (𝑛 − 3) … … 4 × 3 × 2 × 1

GOAL:

• Represent the number of arrangements of 𝑛 objects taken 𝑛 at a time, using factorial notation.

• Determine the number of permutations of 𝑛 objects taken 𝑛 at a time.

• Determine the number of permutations of 𝑛 objects taken 𝑟 at a time.

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You Try: Evaluate the following using the factorial key on a calculator.

a) 4! b) 6!

4! c)

(12!5!)

(2!9!) d)

12!

8!×2! e)

300!

299!

Example 2 There are 8 students waiting in line to get into the Zoo. In how many ways can these students be re-arranged?

Example 3 A grandmother has 9 pictures of her grandchild. She wants to place them in the frame shown below.

How many different ways can she place the 9 photos?

Example 4

Three new teachers are starting at a new school. There are 3 classrooms available. How many ways can

the classrooms be assigned to the new teachers?

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In some situations, we may be only arranging part of a set of objects.

Example 5 There are 15 students special events committee. They are organizing a party. They need one person to get drinks, one to get food, and one to find a location. In how many ways can these roles be assigned?

Example 6 How many four-digit passcodes can be created using the digits 1, 2, 3, 4, 5, 6? Repetition of the digits is not allowed.

Example 7 How many four-digit passcodes can be created using the digits 1, 2, 3, 4, 5, 6? Repetition of the digits is allowed. The number of permutations of “𝑟” objects taken from a group of “𝑛” objects can also be obtained by using the permutation formula: Most scientific calculators include a permutation function, which is usually labelled “nPr”.

𝑃𝑟𝑛 =𝑛!

(𝑛 − 𝑟)!

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Assignment 2 1. How many ways can the letters in the word KINDEST be re-arranged? Write your answer both in

factorial notation and as a normal number. 2. In how many ways can eight people sit in eight different seats at a movie theatre? 3. At a football game they select four lucky people from a row that contains 20 people to win 4

different prizes. In how many ways can the prizes be awarded? 4. How many three-letter permutations can be created out of the letters in the word JOURNEYS? 5. An art teacher has to decide which art projects to place in the display case. She has 35 projects to

choose from but only room for 6 of them. How many different display arrangements are possible? 6. The word UNCOPYRIGHTABLE is one of two longest words in the English language that does not

contain any repeating letters. a) In how many ways can you rearrange ALL of the letters in the word UNCOPYRIGHTABLE? b) How many five-letter arrangements can be made using the letters in the word

UNCOPYRIGHTABLE? 7. Your teacher needs four volunteers from the class: one person to clean the board, one person to

take notes, one person to update the blog and one person to run the projector. If there are 23 people in the class, in how many ways can the tasks be filled?

8. The Salvation Army runs stores that sell gently used goods. They have five spaces in their front

window that they reserve for items that stand out from the rest. How many different window displays are possible if they have 48 items in total?

9. While playing a game, Cathy pulls out the following letters from a bag containing the letters of the

alphabet.:

a) How many different arrangements can be made using all of the letters above?

b) How many different arrangements are possible if only 5 of the above letters are used?

10. Nine children need to line up to get into their classroom at school. If they line up randomly, what is

the probability that Ahmed will be first in line?

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Lesson 3: Permutations Involving Identical Objects

The next type of permutations that need to be explored is permutations of objects where some of the objects are identical.

Example 1 In how many ways can you re-arrange the letters in the word SEA? Once you have determined the number of ways it can be done, list the different ways.

Example 2 In how many ways can you re-arrange the letters in the word SEE? Is this answer different from example 1? List the different possibilities.

GOAL:

• Determine the number of permutations possible when some of the elements are identical.

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Example 3 In how many ways can you rearrange the letters in the word FLUFFY?

Example 4 In how many ways can you re-arrange the letters in the word MISSISSIPPI?

Example 5 A grocer wants to put cans of soup in a line on a shelf. They have six identical cans of tomato soup, four identical cans of chicken noodle soup, seven identical cans of clam chowder, and one can of cream of mushroom. In how many ways can these cans be placed on the shelf?

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Example 6 Using the map below, in how many ways can Jackie go from home to school if she is only allowed to travel North or West?

Example 7 How many direct paths are there from A to B?

School

Home

A

B

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Example 8

Using the diagram below, in how many ways can Jackie go from home to school if she is only allowed to travel North or West and she must pass through the post office on the way there?

Example 9 How many routes are there from start to finish if you only go east and south?

Home

School Post Office

Start

End

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Assignment 3 1. In how many ways can you arrange the letters in the word SUCCESS? 2. Determine how many ways you can arrange the letters in the word EFFERVESCENCE.

3. A bookstore has 14 books they want to put on their ‘FOR SALE’ shelf: Three identical copies of “The

Life of Pi”, six identical copies of “The Time-Traveller’s Wife”, and five identical copies of “Catch-22”. In how many ways can these books be placed in a row on the shelf?

4. How many different six-digit numerals can be written using all of the following six digits: 4, 4, 5, 5, 5, 7?

5. a) How many routes are there from home to the park if you can only travel north and east?

b) How many routes are there from the park to home if you can only travel south and west and you must stop at the store on the way home?

c) In part (b), if a route is randomly chosen, what is the probability that you pass by the store?

home

park

park store

home

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6. How many routes are there from start to finish if you can only travel down and to the right?

7. On the game board shown, the checker is allowed to move toward the top of the board diagonally

left or right. If the X is encountered, the checker cannot move into that square or jump over it.

Determine the number of paths the red checker can follow from its starting position to any white

square along the top of the board.

8. Create a scenario where the solution is 8!

4!3!

start

finish

Top of board

start

XX

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Lesson 4: Permutations Involving Restrictions

Some permutation problems contain special considerations called restrictions. A restriction can take on many forms, but at its simplest level it is a rule that forbids you from putting an item in a certain position in a permutation. Examine the next examples for more.

Example 1 A group of 18 people get together weekly to play board games. This group of people has a leader and a treasurer.

a) In how many ways can the two roles be filled from the 18 people available?

b) In how many ways can the two roles be filled from the 18 people if one of the members refuses to be the leader?

Example 2 How many permutations of the word ORANGE are there that begin with a vowel?

Example 3 How many permutations for the word ORANGE are there that begin AND end with a vowel?

GOAL:

• To determine the number of possible permutations where there are various restrictions on specific positions.

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Example 4 Two parents take 5 children to the movies. In how many ways can they be seated in a row if there must be a parent at both ends?

Example 5 How many three-digit passcodes can be created using digits from 1 through 9, without repetition, if the first digit must be 7?

Example 6 How many four-digit even numbers can be created using the digits 1, 2, 3, 4, and 5, if repetition of digits is not allowed?

Example 7 How many four-digit even numbers can be created using the digits 1, 2, 3, 4, and 5, if repetition of digits is allowed?

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Problems Involving Cases Sometimes choosing items for one restriction location affects the number of choices in another restriction location. In these situations, you must structure the solution into separate cases.

Example 1 How many four character passcodes can be created using the letters A,B, C, D, E, or F if the passcode must start with A or B or C and end with A or F.

Example 2 Eight people will sit in a single line in a canoe. In how many ways can they be seated if Mel or Pat must sit in the front seat, and only Mel, Joe, Gino, or Hamad can sit in the rear seat?

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Assignment 4 1. a) In how many ways can you arrange the letters in the word HOPEFUL so that the arrangements

begin with a consonant and end with a vowel? b) What is the probability that the arrangement begins with a consonant and ends with a vowel?

2. In how many ways can 3 men and 7 women be seated in a row if there must be a woman at the beginning and end of the row?

3. Three parents and 11 children are going to a rock concert. In how many ways can they be seated in a row if there must be a parent in the first seat?

4. A password must be eight characters long, and repetition is allowed. A ‘character’ can be: a lower

case letter (a to z), an upper case letter (A to Z), a number (0 to 9), or one of the following special characters # $ % & * ^ ! a) How many passwords are possible if there are no restrictions? b) How many passwords are possible if you must start with a lower case letter and end with one of

the special characters? 5. A regular license plate in Manitoba contains three capital letters, followed by three digits. Repetition

of letters or digits is allowed. The picture shows you a sample plate:

a) How many Manitoba license plates are possible? b) How many Manitoba license plates are possible if the ‘letters’ section cannot begin with a ‘C’ or

an ‘X’ and the ‘digits’ section cannot start with a 0? 6. How many three-letter words without any repeated letters are possible if the words must start with

an A, B, or C, and end in a vowel? (The letter ‘Y’ is NOT considered a vowel.)

7. A bookstore has eight different books to put in a window display. Each book is a different colour – red, green, blue, orange, pink, yellow, black, and white. How many ways can these books be arranged if the first book in the display must be red, green or blue, and the last book must be pink or blue?

8. How many ways can the letters in the word LEARNS be arranged if you must begin with an A, an L, or

an N, and end with a vowel? 9. Carlos uses the letters of his name to create a passcode. How many four or five letter passcodes are

possible?

10. How many five-digit odd numbers can be created using the digits 1 through 9, where repetition of digits is not allowed?

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Lesson 5: Permutations Involving Groups

In the previous lesson you looked at permutations with some common restrictions. This lesson focusses on a new type of restriction, when some of the items being permuted must stay in a group.

Example 1 In how many ways can you rearrange the letters in the word MAGPIE if the word ‘GAP’ must show up somewhere in the re-arrangement?

Example 2 In how many ways can you rearrange the letters in the word MAGPIE if the letters AGP must remain together?

GOAL:

• To determine the number of permutations possible when some of the objects are grouped.

• To determine the complement of the number of permutations where grouping is involved.

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Example 3 You take eight children to the movies. In how many ways can they be seated in a row if Ahmed, Zinni, and Terry are best friends and insist on sitting together?

Example 4 The University of Winnipeg bookstore has 12 textbooks that they want to display in their store window. There are 3 different science textbooks, 5 different mathematics textbooks, and 4 different engineering textbooks. In how many ways can the textbooks be displayed in a row if the bookstore wants to keep books from the same subject grouped together?

Example 5 A teachers is lining 10 children up for a class photo. a) How many ways can this be done if Dan and Nick, a pair of identical twins, must be kept together? b) How many ways can this be done if Dan and Nick, a pair of identical twins, must be kept apart?

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Example 6 While playing a board game, Cathy pulls out the following letters: W, E, E, A, C, K, E. a) How many different arrangements can be made using all of the letters she pulled out? b) If Cathy places all the vowels together and all the consonants together, how many different

arrangements can be made using all of the letters above?

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Assignment 5 1. Four high school students and eight other people are going on a boat trip. In how many ways can

they line up to board the boat if the students insist on lining up together? 2. Three teachers, two parents, and five children are lining up for the bus. Determine the number of

ways they can line up if… a) there are no restrictions. b) they stand in three groups: the teachers, the parents, and the children. c) the parents insist on standing together. d) the children insist on standing together. e) they stand in order of oldest to youngest. f) what is the probability that they line up oldest to youngest?

3. In how many ways can you arrange the letters of the word MAGNETIC if

a) there are no restrictions? b) the vowels must remain together? c) the consonants must remain together? d) the word TAN must appear in the arrangement? e) what is the probability that the word TAN appears in the arrangement?

4. In how many ways can you arrange the letters of the word ELEPHANT if

a) there are no restrictions? b) the vowels must remain together? c) the consonants must remain together? d) what is the probability that the vowels remain together?

5. Seven friends are going to sit in a row at a sports event. In how many ways can they sit in the row if

two of the friends are not getting along and refuse to sit together?

6. Vincent has an MP3 player that can play songs in a random order. a) How many different ways can a 12-song playlist be arranged, if each song is played only once? b) What is the probability that Vincent’s 3 favourite songs will be played together when he plays the

12-song playlist?

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Lesson 6: Combinations

While a permutation is a selection and ordering of a group of objects, a combination is a selection of a group of objects where the order of the objects is not important. In other words, a combination is a permutation without the ordering of the objects.

For example, when we are selecting people to sit on a committee, inviting people to a party, or when the winning numbers for a lottery are selected, the order of the selection doesn’t matter because there are no specific positions involved.

We can calculate the number of combinations of a selection of objects by using reasoning or by using a function on our calculator.

Example 1 Determine the number of ways that a coach can select 3 athletes from a team of 6 to represent the school at a competition.

Method 1 We can use the fundamental counting principle to determine the number of permutations for this situation and then divide out the “ordering” of the number of objects.

If 3 students must be selected from 6, the number of combinations possible is:

# 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠 =# 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 3 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑓𝑟𝑜𝑚 6

# 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 3 𝑜𝑏𝑗𝑒𝑐𝑡𝑠

= 6×5×4

3! which equals 20.

Method 2

We can use the 𝐶𝑟𝑛 feature on our calculator, where we are selecting 𝑟 objects from a larger group of 𝑛 objects. In this case, because we are selecting 3 students from a larger group of 6, we would enter into

our calculator: 𝐶36 , which also equals 20.

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠 =𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑜𝑟𝑑𝑒𝑟 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡𝑠

GOALS

• Explore strategies for determining the number of combinations of 𝑛 elements taken 𝑟 at a time.

• Determine the number of combinations of 𝑛 elements taken 𝑟 at a time. Solve problems involving combinations.

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Example 2: How many committees of three people can be selected from a class of 20 students?

Example 3: There are 15 engineers and 12 chemists at an oil company.

a) Determine the number of five-person committees that can be formed from this company.

b) Determine the number of five-person committees that can be formed if there must be exactly 3 engineers on the committee.

Example 4: A school principal is selecting students from a group of volunteers to organize a social

event. There are 12 students in a group, 8 students from Class A and 4 students from Class B.

a) How many different groups of 5 students can be created if there are no restrictions? b) How many different groups of 5 students are possible which include at least 1 student from Class B?

Example 5: a) Debbie is planning a party. She has 9 friends. She must choose 5 friends to invite to her party.

How many guest lists are possible?

b) How many guest lists are possible if Julie, one of the 9 friends, refuses to come?

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Assignment 6 1. Determine the number of ways that three students can be chosen from a group of ten. 2. Determine how many ways there are to choose ten-questions for an exam from a test bank

containing 25 questions.

3. To play the Lotto Max lottery, you must choose seven numbers from 1 to 49. To play in the Lotto 649, you must choose six numbers from 1 to 49. To win each jackpot, the numbers chosen must match the numbers drawn by the lottery corporation. a) Determine the number of possible winning combinations there are for each jackpot. b) What is the probability of winning Lotto Max if you purchase 1 ticket?

4. A big family is holding a reunion. The family consists of two great- grandparents, three grandparents,

31 children, 18 grandchildren, and three great grandchildren. a) If the head table seats eight people, how many different combinations of family members could

be seated at the head table? b) How many different combinations could be seated at the head table if great-grandchildren are

not included? c) How many handshakes would take place if every family member at the reunion shook every other

family member’s hand? 5. From a group of 5 workers, determine the number of ways the supervisor can select

a) Exactly 4 workers to the job. b) At least 2 workers to the job.

6. Determine the number of ways you select a set of four science books and three history books from

six different science books and seven different history books. 7. You wish to select a committee of six from 7 women and 5 men.

a) How many ways can this be done if the committee must include exactly 2 of the women and 4 of the men? b) How many committees can be formed if the majority must be women? c) How many committees can be formed if it must include at most 1 man? d) What is the probability that the committee includes at most 1 man?

8. Ali is having a party and he is allowed to invite six of his ten friends. In how many ways can this be done if his best friend, Tim, must be included?

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Lesson 7: Distinguishing between Permutation and Combination Problems

An important objective in this unit is to decide whether a problem involves a permutation of objects or a combination of objects. Once the problem is identified as either a permutation or a combination problem, you can use any of the methods described in the previous lessons to solve the problem.

Example 1 How many ways can three teachers be selected from a group of 25 teachers?

Example 2 How many ways can a CEO, a director, and a manager of adult learning be selected from a group of 25 candidates?

GOALS:

• Distinguish between permutations and combinations. That is, distinguish between selecting and ordering objects and just selecting objects.

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Example 3 If 12 points are arranged in a circle, how many different lines can be formed by connecting two of the points to form a straight line?

Example 4 There are 50 high school students in a small town. The school’s cross-country coach wants to randomly select 9 students to form a team. a) How many different teams can be created? b) The coach rents a van and drives his team to the race. Calculate the number of ways the students can be seated, if there are 9 passenger seats.

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Assignment 7 1. How many ways can you arrange the letters of the word FLOWER?

2. Fifteen volleyball players line up for a team photo.

a) How many ways can this be done? b) How many ways can this be done if the captain must be in the middle of the line?

3. How many groups of 3 toys can a child choose to take on a vacation from a toy box containing 11

toys? 4. A test bank contains a total of 40 questions. In how many ways can a teacher choose eight of these

questions to put on an upcoming test? 5. A class is made up of 25 students; there are 14 math majors in the class and 11 English majors in the

class. How many ways can the teacher select a group of five of the students if the majority must be math majors.

6. How many arrangements are there of the letters of the word MONOTONOUS under each condition?

a) Without restrictions. b) If each arrangement begins with a T. c) If each arrangement begins with an O.

7. A set of 12 encyclopedia is to be arranged alphabetically on a shelf. How many incorrect

arrangements are possible?

8. Determine the number of three-digit numbers that are divisible by five. 9. If eight points are arranged in a circle, how many triangles can be formed by connecting three of the

points to form each triangle? 10. A club has 11 members.

a) How many different two member committees could be formed from this club? b) How many different three member committees could be formed from this club? c) How many ways could a president, treasurer and secretary be chosen from the members?

11. Mr. Wilson has a briefcase with a three-digit combination lock. He is choosing a combination for the

lock and only wants to use his favourite digits, which are 3, 4, 5, 6, and 7. He is not going to use a number more than once. a) How many different combinations are possible? b) How many of these combinations would be odd? c) How many of these combinations would be even?

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12. From a group of 15 people, how many committees can be formed consisting of two, three, or four people?

13. In a league with 8 teams, each team plays the other teams once. How many total games are played?

14. The Cambodian alphabet has 72 letters.

a) How many three-letter arrangements, with no letters repeating, can be created with this alphabet?

b) How many three-letter arrangements can be created if letters may be repeated?

15. Imagine that you have ten light bulbs, of which three are burnt out. a) In how many ways can you randomly select any four of the light bulbs? b) In how many ways can you select two good light bulbs and two burnt out ones?

16. Ten students are asked to break first into a group of two, and then the remaining students into a

group of three, and then the remaining students into a group of five. In how many ways can this be done?

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Lesson 8: Review Assignment 1. In how many ways can the letters in the word KITCHEN be re-arranged if each arrangement must

start with a vowel and end with a consonant?

2. A crate of radio controlled cars contains ten that work and four that are defective. In how many ways can five cars be selected so that three work and two are defective?

3. Simplify without a calculator then evaluate each of the following:

a)14!

12! b)

9!

3!6!

4. From four carpenters and three plumbers a committee of three people is to be chosen:

a) In how many ways can this committee be chosen? b) Find the number of ways that the committee would contain exactly two carpenters. c) Find the number of ways that the committee would contain at least one plumber.

5. In how many ways can eight children be seated in a row at a movie theatre if Billy and Sally insist on

sitting together? 6. Using the diagram below, in how many ways can you get from home to the movie theatre if you can

only travel East or South? 7. A bookstore has ten books that they want to display, but their display shelf can only hold four of the

books. In how many ways can four of the ten books be arranged on the shelf? 8. Find the number of unique permutations that can be created using all of the letters in the word

STATISTICS. 9. A committee of five students is to be selected from a group of 14 students.

a) In how many ways can this be done? b) In how many ways can this be done if two of the 14 students MUST be on the committee? c) In how many ways can this be done if two of the 14 students CANNOT be on the committee?

Movie Theatre

Home

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10. Six blondes and four brunettes sit in a row. In how many ways can this be done if there must be a blonde at each end of the row?

11. You are starting a stamp collection. You have five Canadian stamps, three American Stamps, four

European stamps and six stamps from China. In how many ways can these stamps be placed in a line if the stamps from each region must stay together?

12. A car dealership has vehicles with the following options: four, six or eight cylinders; red, green or

blue coloring; front wheel drive, rear wheel drive or four wheel drive. a) How many different variety of cars (cylinders, colour, and drive) are sold by the dealership? b) If Gina wants a car from this dealership, but she definitely wants to avoid buying a green car, how

many options are available to her? 13. A bag contains six identical red elastic bands, three identical blue elastic bands and one white elastic

band. If you pull all of the elastic bands out of the bag and place them in a row, how many distinct arrangements can be made?

14. In how many ways can you re-arrange the letters in the word FRAGILE if the word ‘LEG’ must appear

in that order in the re-arrangement? 15. In how many ways can four volunteers be chosen from a group of 25 people?

16. Ella is having her birthday party at a movie theatre. Including Ella, there are 10 kids in total (including

Ella’s brother Ben) at the movie. In how many ways can these 10 kids sit in a row if Ella refuses to sit beside her brother?

17. How many football games would be played if each of the nine teams in a league plays each of the

other teams in that league once? 18. There are five main roads between cities Awesometown and Brilliantville and four main roads

between Brilliantville and Candyland. In how many ways can a person drive from Awesometown to Candyland and return, going through Brilliantville on both trips and NOT driving on the same road twice?

19. How many arrangements of four different letters can be made from the letters

a, e, i, o, r, s, t if: a) there are no restrictions? b) the arrangements must begin and end with a vowel?

20. How many four digit numbers greater than 5000 are possible using the digits

1, 2, 3, 5, 7, and 8 if no repetition of digits is allowed? 21. In how many ways can 5 books be arranged on a book shelf?

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22. In how many ways can a president, vice-president and treasurer be chosen from 23 students? 23. How many different arrangements of the letters in FRIGHTEN can be made if:

a) there are no restrictions? b) the first letter must be a G? c) all arrangements must end in TH? d) the arrangements must begin and end with a consonant?

24. In how many ways can a group of four policemen be chosen for special duty from a group of 12

policemen? 25. In a class of 30 students, each student shakes hands with each of the other students once. How

many handshakes take place? 26. On a Math exam, students must answer 5 of the first 6 questions and 3 of the last 5 questions. In

how many ways can this be done? 27. In how many ways can a committee of five people be chosen from a group of ten brunettes and eight

redheads so that the redheads have a majority on the committee?

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Counting Methods Answer Keys

Assignment 1 1. a) 6 b) GB, GW, YB, YW, OB, OW 2. a) 8 b) RL, RC, BL, BC, WL, WC, SL, SC 3. 20 4. 120 5. 1440 6. 32 7.81 8. a) 8 b) 24 9. a) 4 b) 3 c) 24

10. 2

25

Assignment 2 1. 7! = 5040

2. 40320

3. 116 280

4. 336

5. 1 168 675 200

6. a) 1 307 674 368 000 or 1.31E12 b) 360 360

7. 212520

8. 205 476 480 9. a) 3 628 800 b)30240

10. 40 320

362 880

Assignment 3 1. 420 2. 12 972 960 3. 168 168 4. 60

5. a) 330 b) 140 c) 140

330

6. 146 7. 8 8. Please be prepared to share and justify your scenario.

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Assignment 4

1. a) 1440 b) 1440

5040

2. 1 693 440 3. 1.86810624E10 4. a) 513 798 374 400 000 or 5.14E14 b) 19 641 105 680 000 or 1.96E13 5. a) 17 576 000 b) 14 601 600 6. 336 7. 3600 8. 120 9. 1080 10. 8400

Assignment 5 1. 8 709 120

2. a) 3 628 800 b) 8640 c) 725 760 d) 86 400 e) 1 f) 1

3 628 800

3. a) 40 320 b) 4320 c) 2880 d) 720 e) 720

40 320

4. a) 20 160 b) 2160 c) 1440 d) 2160

20 160

5. 3600

6. a) 479 001 600 b) 21 772 800

479 001 600

Assignment 6 1. 120 2. 3 268 760

3. a) Lotto Max: 85 900 584 Lotto 6/49: 13 983 816 b) 1

85 900 584

4. a) 1 652 411 475 b) 1 040 465 790 c) 1596 5. a) 5 b) 26 6. 525

7. a) 105 b) 462 c) 122 d) 122

924

8. 126

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Assignment 7 1. 720 2. a) 1 307 674 368 000 or 1.31E12 b) 87 178 291 200 or 8.72E10 3. 165 4. 76 904 685 5. 33 033 6. a) 75600 b) 7560 c) 30 240 7. 479 001 599 8. 180 9. 56 10. a) 55 b) 165 c) 990 11. a) 60 b) 36 c) 24 12. 1925 13. 28 14. a) 357 840 b) 373 248 15. a) 210 b) 63 16. 2520

Assignment 8 1. 1200 2. 720 3. a) 182 b) 84 4. a) 35 b) 18 c) 31 5. 10080 6. 210 7. 5040 8. 50400 9. a) 2002 b) 220 c) 1782 10. 1 209 600 11. 298 598 400 12. a) 27 b) 18 13. 840 14. 120 15. 12650 16. 2 903 040 17. 36 18. 240 19. a) 840 b) 240 20. 180 21. 120 22. 10626 23. a) 40320 b) 5040 c) 720 d) 21600 24. 495 25. 435 26. 60 27. 3276

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Counting Methods Summary

These are the outcomes that have been covered in this unit. Check off each box if you are confident that you can demonstrate that skill:

At the end of this Unit, I can:

Represent and solve counting problems using a graphic organizer.

Understand the fundamental counting principle (FCP).

Solve counting problems.

Use the FCP to determine the number of permutations possible.

Determine the number of permutations possible when some elements are identical.

Determine the number of permutations when there are restrictions.

Determine the number of permutations when some objects are grouped.

Determine the complement of a permutation problem.

Determine the number of combinations possible.

Distinguish between permutations and combinations.

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Ongoing Self-Assessment for Mathematics Students

Understanding How confident are you in your ability to demonstrate understanding of the outcomes of this unit? My ability to demonstrate understanding is a: STRENGTH CHALLENGE

Attendance Did you have consistently good attendance during this unit?

My attendance is a: STRENGTH CHALLENGE

Out of Class Practice Did you feel that when you needed to practice a math skill outside of class, you were able to do so?

My ability to practice outside of class time is a: STRENGTH CHALLENGE

Accessing Help If you answered CHALLENGE to any of the questions above, consider the following options for accessing help in order to be more successful in this course:

• Talk to your TEACHER.

• Make time to visit the RESOURCE ROOM (ROOM 104).

• Get help / support / materials from a CLASSMATE.

• Use any resources provided on a CLASS BLOG (if available). You have completed a unit in this Math course. Please take some time to reflect on your thoughts regarding your academic strengths and challenges as they relate to the outcomes of this unit. You can also reflect on any previous outcomes of this course. __________________________________________________________________ __________________________________________________________________