Unit 3 – Polynomial Functions, Equations and Inequalitiesmseko.weebly.com/uploads/1/4/9/9/14997226/chapter_8... · Web viewChapter 8 – Permutations and Combinations Name: _____

Chapter 8 – Permutations and Combinations

Name: _______________

Chapter 8 - Permutations and Combinations ● 1

8.1 The Fundamental Counting Principle Date:Investigate: Counting without Counting1. A café has a lunch special consisting of an egg or a ham sandwich (E or H); milk, juice, or coffee (M, J,

or C); and yogurt or pie for dessert (Y or P).a) One item is chosen from each category. List all possible meals

b) How many possible meals are there?

c) How can you determine the number of possible meals without listing all of them?

The Fundamental Counting PrincipleIf one item can be selected in m ways, and for each way a second item can be selected in n ways, then the two items can be selected in ____________ ways.

Example 1: A computer store sells 5 different computers, 3 different monitors, 5 different printers, and 2 different multimedia packages. How many different computer systems are available?

Example 2: How many even natural numbers are there with 3 digits?

Example 3: In each case, how many different 2-digit numbers are there?

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2 ● Chapter 8 – Permutations and Combinations

a) Repetitions are allowed.

b) Repetitions are not allowed

Example 4: A multiple-choice test has 7 questions, with 4 possible answers for each question. Suppose students answer each question by guessing randomly.

a) How many possible answers are there for each question?

b) How many different patterns are possible for the answers to the 7 questions on the test?

c) What is the probability that all 7 questions will be answered correctly?

Example 5: Using the word ENGLISH, how many distinguishable 4 letter arrangements can be formed if:a) letters can be repeated?

b) no letters repeated and the first letter must be E?

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Example 6: How many license plates can be made with 4 characters in which at least the first 2 of the characters are letters and the rest are digits?

Example 7: How many ways are there of getting from A to C in each diagram passing through each point at most once?

a)

b)

Assignment: page 689- 692 #3-10, 12; page 694 #1, 2

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A B C

A B2 C

B1

B3


8.2 Permutations of Different Objects Date:Investigate:1. Two letters, A and B, can be written in two different orders, AB and BA.

These are permutations of A and B.a) List all of the permutations of 3 letters A, B, and C.

How many permutations are there?

b) List all of the permutations of 4 letters A, B, C, and D.

How many permutations are there?

c) Predict the number of permutations of 5 letters A, B, C, D, and E.

Instead of arranging letters in order, we can arrange objects if they are all different.

Example 1: When you press the “scramble” button on a CD player it plays a permutation of the songs on the CD. If the CD has 5 songs on it, how many permutations of the songs are possible?

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Factorial Notation The symbol ! is used in mathematics to denote the factorial operation.

1! =2! =3! =4! =5! =

n! =

Permutations An _________________ arrangement of ______________________ is called a permutation.

The number of permutations of n distinct objects is .

The number of permutations of n distinct objects taken r at a time is .

Notation: 37 P

Example 2: How many 3-letter permutations can be formed using the letters of the word COMPUTE?

Example 3: From a group of 100 people, how many ways can a president, vice-president, and treasurer be selected?

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Definition of 0!

0! = Why?

Practice:1. Write each expression without using the factorial symbol.

a)

b)

c)

2. Find the values of the following:

a) b) c)

3. Simplify each expression:

a) b) c)

d)

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4. Write each expression without using the factorial symbol:

a) b)

5: Solve each equation for n.

a) b)

c) d)

6. How many ways can 4 boys and 2 girls sit in a row if:

a) there are no restrictions of where they sit in the row?

b) all the boys and all the girls sit together?

Assignment: page 702-705 #4-10; page 706 #1-3

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8.3 Permutations Involving Identical Objects Date:

Permutations with Identical ObjectsThe number of permutations of n objects of which there are a objects alike of one kind, b alike of another kind, c alike of another kind, and so on, is

Example 1: Determine the number of permutations of all the letters in each of the following words.

a) OGOPOGO

b) STATISTICIAN

Example 2: A true-false test has 7 questions. How many answer keys are possible if 3 answers are T and 4 answers are F?

Example 3: Chris, Sandy, Kurtis, Theresa and Andrea go to watch a movie and sit in 5 adjacent seats. In how many ways can this be done under each condition?

a) without restriction

b) if Chris sits next to Sandy

c) if Sandy refuses to sit next to Chris

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Example 4: In how many ways can four adults and five children be arranged in a single line under each condition?

a) without restrictions

b) if children and adults are alternated

c) if the adults are all together and the children are all together

d) if the adults are all together

Example 5: On the following grid, how many different paths can A take to get to B, assuming one can only travel east and south? Explain.


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A

B


8.4 Combinations Date:

Investigate:1. If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place,

be awarded?

Does the order of finish for the fastest three matter here?

This is an example of a permutation of objects taken at a time.

2. If 5 sprinters compete in a race and the fastest 3 qualify for the relay team, how many different relay teams can be formed?

Visualize the 5 sprinters below. Since 3 will qualify for the relay team and 2 will not, consider the number of ways of arranging 3 Y’s and 2 N’s.

Y Y Y N N

Does the order of finish for the fastest three matter here?

This is an example of a combination of objects taken at a time.

Combinations An unordered arrangement of distinct objects is called a combination.

The number of combinations of n distinct objects taken r at a time is .

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Example 1:a) How many different committees of 3 people can be formed from 7 people?

b) How many different committees of 3 people can be formed from 7 people if the first person selected serves as the chairperson, the second as the treasurer, and the third as the secretary?

c) If the group of 7 people consists of 3 males and 4 females, how many different committees of 3 people can be formed with 1 male and 2 females?Think: you must choose 1 male out of the group of 3 males and 2 females out of the group of 4 females.

d) If the group of 7 people consists of 3 males and 4 females, how many different committees of 3 people can be formed with at least one male on the committee?

Example 2:

a) Evaluate: b) Solve:

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Example 3: A standard deck of 52 playing cards consists of 4 suits (spaces, hearts, diamonds, and clubs) of 13 cards each.

a) How many different 5-card hands can be formed?

b) How many different 5-card hands can be formed that consist of all hearts?

c) How many different 5-card hands can be formed that consist of all face cards?

d) How many different 5-card hands can be formed that consist of 3 hearts and 2 spades?

e) How many different 5-card hands can be formed that consist of exactly 3 hearts?

f) How many different 5-card hands can be formed that consist of at least 3 hearts?

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Club Diamond Heart Spade

KQJ1098765432A


Example 4: A basketball coach has five guards and seven forwards on his basketball team.a) In how many different ways can he select a starting line-up of two guards and three forwards?

b) How many starting teams are there if the star player, who plays guard, must be included?

Example 5:a) Use factorial notation to show

b) Explain in words why is the same as .

c) Prove the identity

Example 6: During a basketball tournament, all players shake hands with each other at the end of the last game. If 300 handshakes were exchanged, how many were at the tournament?

Mixing it up!

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a) If there are 8 horses in a race, how many different ways can 3 horses advance to the next round?

b) If there are 8 horses in a race, how many different ways can 3 horses be awarded 1st, 2nd, and 3rd place?

c) Calculate the number of ways a president, vice president and a treasurer be selected from a class of 30 students.

d) Calculate the number of ways a group of 3 people be selected for student council from a class of 30 students.

e) How many 3-letter words are there of the word SAMPLE?

f) How many ways are there for all the letters in PARALLEL?

g) A multiple choice test has 8 questions, with 4 possible answers for each question. Suppose the answer to each question is a guess. How many different ways are there to complete the test?

h) A multiple choice test has 8 questions, with 3 answers are A, 2 answers are B, 2 answers are C, and 1 answer is D. How many different answer keys are possible?


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8.5 Pascal’s Triangle Date:Generation of Pascal’s Triangle

Observations:1. Relationship to

2. The Symmetrical Pattern

Justification

Generalization

3. The Recursive Pattern

Justification

Generalization

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Pathway Problems Revisited

1. On each grid, how many different paths can A take to get to B, assuming that you can only move to the right and down?

a) b )

2. A rook is a chess piece that can only move horizontally or vertically on a chessboard. A rook is positioned on the northwest corner of a chessboard. How many different paths exist for the rook to move to the southeast corner if the rook moves eastward and southward only?

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A

B

A

B


3. How many different paths are there from A to B, assuming that you are always moving closer to B?

Assignment: page 737 #1-4 and worksheet

8.6 Permutations Involving Identical Objects Date:

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A

B

A

B


Investigate the Patterns in Binomial Powers:a) Expand and simplify each of the following powers of the binomial a + b:

b) What pattern do you see in the

powers of a?

powers of b?

numerical coefficients in the expansions?

c) Confirm your observations about the patterns by expanding and simplifying

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Visualizing – The Binomial Expansion of (a + b)4

0 b’s 1 b 2 b’s 3 b’s 4 b’sNumber of ways to choose this many b’s from 4 factors of (a + b)

Example 1: Expand

a)

b)

c)

The Binomial Theorem (using combinations)

For any whole number n:

(a + b)n = nC0 anb0 + nC1 an-1b1 + nC2 a n-2b2 + … + nCk an-kbk + … + nCn a0bn

1st term 2nd term 3rd term term (n+1)th term

For this expansion the (k + 1)th term is: .

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Expand:e)

f)

Example 2: Determine the 4th term in the expansion of .

Example 3: Write the first four terms of the binomial expansion of .

Example 4: Find the term in x3 in .

Example 5: One term in the expansion of is 448x6. Determine the value of a.

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Example 6: The 4th term in the expression of is . Determine n.

Example 7: Determine the constant term in the expansion of .

Assignment: page 743-749 #3-11, 13, 16; page 750 #1-3Chapter Review: page 719 -720 #1-10; page 754-758 #1-15; page 759-760 #1-8Final Exam Review: page 761-772 #1-24 and Practice Final Package

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Unit 3 – Polynomial Functions, Equations and Inequalitiesmseko.weebly.com/uploads/1/4/9/9/14997226/chapter_8... · Web viewChapter 8 – Permutations and Combinations Name: _____