chapter 9

Chapter 9

Discrete Mathematics

9.1Basic Combinatorics

Quick Review

Give the number of objects described.1. The number of cards in a standard deck.2. The number of face cards in a standard deck.3. The number of vertices of a octogon.4. The number of faces on a cubical die.5. The number of possible totals when two dice are rolled.

Quick Review Solutions

Give the number of objects described.1. The number of cards in a standard deck. 2. The number of face cards in a standard deck. 3. The number of vertices of a octogon. 4. The number of fac

8es on a cubical die.

5. The number of possible totals when two dice are rolled6

What you’ll learn about

Discrete Versus Continuous The Importance of Counting The Multiplication Principle of Counting Permutations Combinations Subsets of an n-Set

… and whyCounting large sets is easy if you know the correct formula.

Multiplication Principle of Counting

If a procedure has a sequence of stages , ,..., and if can occur in ways, can occur in ways

can occur in ways,then the number of ways that the procedure can occur is the produ

P S S SS rS r

1 2ct ... .

Example Using the Multiplication Principle

If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.

Example Using the Multiplication Principle

If a license plate has four letters followed by three numerical digits. Find the number of different license plates that could be formed if there is no restriction on the letters or digits that can be used.

You can fill in the first blank 26 ways, the second blank 26 ways, the third blank 26 ways, the fourth blank 26 ways, the fifth blank 10 ways, the sixth blank 10 ways, and the seventh blank 10 ways. By the Multiplication Principle, there are 26×26×26×26×10×10×10 = 456,976,000 possible license plates.

Permutations of an n-Set

There are n! permutations of an n-set.

Example Distinguishable Permutations

Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

Example Distinguishable Permutations

Count the number of different 8-letter “words” that can be formed using the letters in the word COMPUTER.

Each permutation of the 8 letters forms a different word. There are 8! = 40,320 such permutations.

Distinguishable Permutations

There are ! distinguishable permutations of an -set containing distinguishable objects.If an -set contains objects of a first kind, objects of a secondkind, and so on, with ... , t

n n nn n n n

hen the number of !distinguishable permutations of the -set is .

! ! ! !k

nnn n n n

Permutations Counting Formula

The number of permutations of objects taken at a time is !denoted and is given by for 0 .

If , then 0.

n r n r

n rnP P r n

Combination Counting Formula

The number of combinations of objects taken at a time is !denoted and is given by for 0 .

If , then 0.

n r n r

n rnC C r n

Example Counting Combinations

How many 10 person committees can be formed from a group of 20 people?

Example Counting Combinations

How many 10 person committees can be formed from a group of 20 people?

Notice that order is not important. Using combinations,20! 184,756.

10! 20 10 !

There are 184,756 possible committees.

Formula for Counting Subsets of an n-Set

There are 2 subsets of a set with objects (including theempty set and the entire set).

9.2The Binomial Theorem

Quick Review

Use the distributive property to expand the binomial.

2. ( 2 )3. (2 3 )4. (2 )

a bc dx y

3 2 2 3

Use the distributive property to expand the binomial.

2. ( 2 ) 3. (2 3 )

4. (2 )

12 94 4

a bc dx

x xy y

a ab bc cd d

x xy y

x x y y yy x

Powers of Binomials Pascal’s Triangle The Binomial Theorem Factorial Identities

… and whyThe Binomial Theorem is a marvelous study in combinatorial patterns.

Binomial Coefficient

The binomial coefficients that appear in the expansion of ( ) are the values of for 0,1,2,3,..., .A classical notation for , especially in the context of binomial

coefficients, is .

a bC r n

Both notations are read " choose ."n r

Example Using nCr to Expand a Binomial

Expand , using a calculator to compute the binomial coefficients.a b

Example Using nCr to Expand a Binomial

Expand , using a calculator to compute the binomial coefficients.a b

4 4 3 2 2 3

Enter 4 0,1, 2,3,4 into the calculator to find the binomial

coefficients for 4. The calculator returns the list 1,4,6,4,1 .

Using these coefficients, construct the expansion:

a b a a b a b ab

Recursion Formula for Pascal’s Triangle

1 1 or, equivalently,

1 n r n r n r

n n nC C C

The Binomial Theorem

For any positive integer ,

... ... ,0 1

!where .!( )!

n n n n r r n

nn n n n

a b a a b a b br n

n nCr r n r

Basic Factorial Identities

For any integer 1, ! 1 !

For any integer 0, 1 ! 1 !

n n n n

9.3Probability

Quick Review

How many outcomes are possible for the following experiments.1. Two coins are tossed.2. Two different 6-sided dice are rolled.3. Two chips are drawn simultaneously without replacement froma jar with 8

chips.4. Two different cards are drawn from a standard deck of 52.

5. Evaluate without using a calculator. CC

How many outcomes are possible for the following experiments.1. Two coins are tossed. 2. Two different 6-sided dice are rolled. 3. Two chips are drawn simultaneously without replacement froma

with 8 chips. 4. Two different cards are drawn from a standard deck of 52.

5. Evaluate without using a calculat

281326

or. 14 3/CC

Sample Spaces and Probability Functions Determining Probabilities Venn Diagrams and Tree Diagrams Conditional Probability Binomial Distributions

… and whyEveryone should know how mathematical the “laws of chance” really are.

Probability of an Event (Equally Likely Outcomes)

If is an event in a finite, nonempty sample space of equally likely outcomes, then the of the event is

the number of outcomes in ( ) .the number of outcomes in

probability

Probability Distribution for the Sum of Two Fair Dice

Outcome Probability2 1/363 2/364 3/365 4/366 5/367 6/368 5/369 4/3610 3/3611 2/3612 1/36

Example Rolling the Dice

Find the probability of rolling a sum divisible by 4 on a singleroll of two fair dice.

Example Rolling the Dice

Find the probability of rolling a sum divisible by 4 on a singleroll of two fair dice.

The event consists of the outcomes 4,8,12 . To get the probability

of we add up the probabilities of the outcomes in :3 5 1 9 1

( ) .36 36 36 36 4

Probability Function

A is a function that assigns a real numberto each outcome in a sample space subject to the following conditions:1. 0 ( ) 1;2. the sum of the probabilities of all outcomes in

probability function

is 1;3. ( ) 0.

Probability of an Event (Outcomes not Equally Likely)

Let be a finite, nonempty sample space in which every outcomehas a probability assigned to it by a probability function . If isany event in , the of the event is the sum of the pro

S Eprobabilitybabilities of all the outcomes contained in . E

Strategy for Determining Probabilities

1. Determine the sample space of all possible outcomes. When possible,choose outcomes that are equally likely.2. If the sample space has equally likely outcomes, the probability of an

event is determE the number of outcomes in ined by counting: ( ) .the number of outcomes in

3. If the sample space does not have equally likely outcomes, determinethe probability function. (This is not always easy

to do.) Check to be surethat the conditions of a probability function are satisfied. Then the probability of an event is determined by adding up the probabilitiesof all the outcomes contained in .

Example Choosing Chocolates

Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

The experiment in question is the selection of three chocolates,without regard to order, from a box of 12. There are 220outcomes of this experiment. The event E consists of all possiblecombinat

ions of 3 that can be chosen, without regard to order, fromthe 5 vanilla cremes available. There are 10 ways.Therefore, ( ) 10 / 220 1/ 22.

Multiplication Principle of Probability

Suppose an event A has probability p1 and an event B has probability p2 under the assumption that A occurs. Then the probability that both A and B occur is p1p2.

The probability of picking a vanilla creme on the first draw is 5/12.Under the assumption that a vanilla creme was selected in the firstdraw, the probability of picking a vanilla creme on the second draw is 4/11. Under the assumption that a vanilla creme was selected in the first and second draw, the probability of picking a vanilla creme on the thirddraw is 3/10. By the Multiplication Principle, the probability of picking

5 4 3 60 1a vanilla creme on all three picks is .12 11 10 1320 22

Conditional Probability Formula

( and )If the event depends on the event , then ( | ) .( )

P A BB A P B AP A

Binomial DistributionSuppose an experiment consists of -independent repetitions of an experiment with two outcomes, called "success" and "failure." Let

(success) and (failure) . (Note that 1 .)Then the terms in th

P p P q q p e binomial expansion of ( ) give the respective

probabilities of exactly , 1,..., 2, 1, 0 successes.

np qn n

Number of successes out of Probability independent repetitionsn

n p qn

Example Shooting Free Throws

Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

(15 successes) 0.92 0.286P

Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

10 515(10 successes)= 0.92 0.08 0.00427

9.4Sequences

Quick Review

Evaluate each expression when 3, 2, 4 and 2.1. ( 1) 2. Find .

4. 2 35. 3 and 10

a r n da n da r

kaa a a

Evaluate each expression when 3, 2, 4 and 2.1. ( 1) 2. Find .

4. 2 3

39,36 5. 3 and 10

a r n da n da r

kaa a a

Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences Sequences and Graphing Calculators

… and whyInfinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

Limit of a Sequence

Let be a sequence of real numbers, and consider lim .

If the limit is a finite number , the sequence and is the . If the limit is infinite or nonexistent,the se

n nna a

convergeslimit of the sequencequence .diverges

Example Finding Limits of Sequences

Determine whether the sequence converges or diverges. If it converges,give the limit.

2 1 2 22,1, , , ,..., ,...3 2 5 n

Example Finding Limits of Sequences

Determine whether the sequence converges or diverges. If it converges,give the limit.

2 1 2 22,1, , , ,..., ,...3 2 5 n

2lim 0, so the sequence converges to a limit of 0.n n

Arithmetic Sequence

A sequence is an if it can be written in the

form , , 2 ,..., ( 1) ,... for some constant .

The number is called the .Each term in an arithmetic seque

a a d a d a n d d

arithmetic sequence

common difference

nce can be obtained recursively fromits preceding term by adding : (for all 2).

n nd a a d n

Example Arithmetic Sequences

Find (a) the common difference, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.

-2, 1, 4, 7, …

Example Arithmetic Sequences

Find (a) the common difference, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.

-2, 1, 4, 7, …

(a) The common difference is 3.(b) 2 (10 1)3 25(c) 2 3 for all 2(d) 2 3( 1) 3 5

aa a a na n n

Geometric Sequence

A sequence is a if it can be written in the

form , , ,..., ,... for some nonzero constant .

The number is called the .Each term in a geometric sequence

a a r a r a r r

geometric sequence

common ratio

can be obtained recursively fromits preceding term by multiplying by : (for all 2).

n nr a a r n

Example Defining Geometric Sequences

Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.

2, 6, 18,…

Example Defining Geometric Sequences

Find (a) the common ratio, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.

2, 6, 18,…

(a) The ratio is 3.(b) 2 3 39,366(c) 2 and 3 for 2.(d) 2 3 .

aa a a na

Sequences and Graphing Calculators

One way to graph a explicitly defined sequences is as scatter plots of the points of the form (k,ak).

A second way is to use the sequence mode on a graphing calculator.

The Fibonacci Sequence

The Fibonacci sequences can be defined recursively by11

for all positive integers 3.n n n

aaa a a

9.5Series

Quick Review

is an arithmetic sequence. Use the given information to find .

1. 5; 42. 5; 2

is a geometric sequence. Use the given information to find .

3. 5; 44. 5; 4

5. Find the sum of

a da d

a ra r

2 the first 3 terms of the sequence .n

is an arithmetic sequence. Use the given information to find .

1. 5; 4 2. 5; 2

is a geometric sequence. Use the given information to find .

1,310,7205; 4 4.

a da d

5; 4 5. Find the sum of the first 3 terms of the sequence

81,92014.

Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series

… and whyInfinite series are at the heart of integral calculus.

Summation Notation

In , the sum of the terms of the sequence , ,...,

is denoted which is read "the sum of from 1 to ."

The variable is called the .

a a k n

summation notation

index of summation

Sum of a Finite Arithmetic Sequence

Let , ,..., be a finite arithmetic sequence with common difference .

Then the sum of the terms of the sequence is

2 ( 1)2

a a a d

a a a a

n a n d

Example Summing the Terms of an Arithmetic Sequence

A corner section of a stadium has 6 seats along the front row. Eachsuccessive row has 3 more seats than the row preceding it. If the toprow has 24 seats, how many seats are in the entire section?

Example Summing the Terms of an Arithmetic Sequence

A corner section of a stadium has 6 seats along the front row. Eachsuccessive row has 3 more seats than the row preceding it. If the toprow has 24 seats, how many seats are in the entire section?

The number of seats in the rows form an arithmetic sequence with6, 24, and 3. Solving

( 1)24 6 3( 1)

7Apply the Sum of a Finite Sequence Theorem:

6 24Sum of chairs 7 105. T2

a a da a n d

here are 105 seats in the section.

Sum of a Finite Geometric Sequence

Let , ,..., be a finite geometric sequence with common ratio .

Then the sum of the terms of the sequence is

a a a r

a a a a

Infinite Series

An infinite series is an expression of the form ... ...k nk

a a a a

Sum of an Infinite Geometric Series

1The geometric series converges if and only if | | 1.

If it does converge, the sum is .1

ka r r

Example Summing Infinite Geometric Series

Determine whether the series converges. If it converges, give the sum.

2 0.25k

Example Summing Infinite Geometric Series

Determine whether the series converges. If it converges, give the sum.

2 0.25k

Since | | 0.25 1, the series converges.2 8The sum is .

1 1 0.25 3

9.6Mathematical Induction

Quick Review

1. Expand the product ( 2)( 4).Factor the polynomial.2. 7 103. 3 3 1

4. Find ( ) given ( ) .1

5. Find ( 1) given ( ) 1.

n nn n n

xf t f xx

f t f x x

1. Expand the product ( 2)( 4). Factor the polynomial.2. 7 10

3. 3 3 1

4. Find ( ) given ( ) . 1

5. Find ( 1) given ( ) 1.

xf t f xx

f t f x

The Tower of Hanoi Problem Principle of Mathematical Induction Induction and Deduction

… and whyThe principle of mathematical induction is a valuable technique for proving combinatorial formulas.

The Tower of Hanoi Solution

The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2n – 1.

Principle of Mathematical Induction

Let Pn be a statement about the integer n. Then Pn is true for all positive integers n provided the following conditions are satisfied:1. (the anchor) P1 is true;2. (inductive step) if Pk is true, then Pk+1 is

9.7Statistics and Data (Graphical)

Quick Review

Solve for the require value.1. 567 is what percent of 12345?2. 73 is what percent of 360 ?3. 357 is 35.7% of what number?Round the given value to the nearest whole number in the specified units.4. 12

34 millions (billions)5. 1,234,567 (millions)

Solve for the require value.1. 567 is what percent of 12345? 2. 73 is what percent of 360 ?

4.593%20.278%

3. 357 is 35.7% of what number? Round the given value to the nearest whole number in

the specified units.4. 1234 millions (billions) 5. 1,234,567 (millions)

1 billion1 million

Statistics Displaying Categorical Data Stemplots Frequency Tables Histograms Time Plots

… and whyGraphical displays of data are increasingly prevalent in professional and popular media. We all need to understand them.

Leading Causes of Death in the United States in 2001

Cause of Death Number of Deaths Percentage

Heart Disease 700,142 29.0Cancer 553,768 22.9Stroke 163,538 6.8Other 1,018,977 41.3

Source: National Center for Health Statistics, as reported in The World Almanac and Book of Facts 2005.

Bar Chart, Pie Chart, Circle Graph

Example Making a Stemplot

Make a stemplot for the given data.

12.323.412.024.523.718.722.419.524.524.6

Example Making a Stemplot

Make a stemplot for the given data.

12.323.412.024.523.718.722.419.524.524.6

Stem Leaf

12 0,3

23 4,7

24 5,5,6

Time Plot

9.8Statistics and Data (Algebraic)

Quick Review

2 2 3 3 4 4 5 5

2 2 2 2

1 2 3 4

Write the sum in expanded form.

12. 313. 3

Write the sum in sigma notation4.

15. 20

x f x f x f x f

x x x x x x x x

2 2 3 3 4 4 5 5

0 1 2 3 4 5

Write the sum in expanded form.

12. 313. 3

Write the sum in sigma notation

15. 20

x f x f x f x f

x x x x x x

13 4 1

iixx x x x xx x

Parameters and Statistics Mean, Median, and Mode The Five-Number Summary Boxplots Variance and Standard Deviation Normal Distributions

… and whyThe language of statistics is becoming more commonplace in our everyday world.

The of a list of numbers , ,..., is

... 1 .

n x x x

x x xx x

Median

The median of a list of n numbers {x1,x2,…,xn} arranged in order (either ascending or descending) is the middle number if n is odd, and the mean of the two middle numbers if n is

The mode of a list of numbers is the number that appears most frequently in the list.

Example Finding Mean, Median, and Mode

Find the (a) mean, (b) median, and (c) mode of the data: 3, 6, 5, 7, 8, 10, 6, 2, 4, 6

Example Finding Mean, Median, and Mode

Find the (a) mean, (b) median, and (c) mode of the data: 3, 6, 5, 7, 8, 10, 6, 2, 4, 6

3 6 5 7 8 10 6 2 4 6(a) 5.710

(b) Put the data in order: 2, 3, 4, 5, 6, 6, 6, 7, 8, 10.The median is 6.(c) The mode is 6.

Weighted Mean

1 1 2 2 1

The formula for finding the mean of a list of numbers , ,..., with

...frequencies , ,..., is .

i in n i

x fx f x f x ff f f x

f f f f

Five-Number Summary

The of a data set is the collectionminimum, , median, , maximum .Q Q

five - number summary

Boxplot

Outlier

A number in a data set can be considered an outlier if it is more than 1.5×IQR below the first quartile or above the third quartile.

Standard Deviation

The of the numbers , ,..., is

1 , where denotes the mean.

The is , the square of the standard deviation.

x x xn

standard deviation

variance

Normal Curve

The 68-95-99.7 Rule

If the data for a population are normally distributed with mean μ and standard deviation σ, then Approximately 68% of the data lie between μ - 1σ

and μ + 1σ. Approximately 95% of the data lie between μ - 2σ

and μ + 2σ. Approximately 99.7% of the data lie between μ - 3σ

and μ + 3σ.

The 68-95-99.7 Rule

Chapter Test

1. A travel agent is trying to schedule a client's trip from city Ato city B. There are three direct flights, three flights from A toa connecting city C, and four flights from this connecting cityC t

o city B. How many trips are possible?2. A club has 45 members, and its membership committee hasthree members. How many different membership committeesare possible?

3. Find the coefficient of in tx y 8

he expansion of 2 .

4. List the elements in the sample space. A game spinner on a circular region divided into 6 equal sectors numbered 1 - 6 is spun.

Chapter Test

5. A fair coin is tossed four times. Find the probability of obtainingone head and three tails.6. Two cans of mixed nuts of different brands are open on a table.Brand A consists of 30% cashews, while brand B consists of 40%cashews. A can is chosen at random, and a nut is chosen at randomfrom the can. Find the probability that the nut is(a) from the brand A can.(b) a brand A cashew.(c) a cashew.(d) from the brand A can, given that it is a cashew.

Chapter Test

7. Find the first 6 terms and the 12th term of the sequence given5 and 2 , for 2.

8.Find an explicit formula for the th term of the arithmetic sequence5, 1,3,7,...

9. Find the sum of the te

k kb b b k

1 1 1rms of the geometric sequence 3, 1, , ,3 9 27

10. Determine whether the geometric series 3 0.5 converges. If it does,

find its sum.

Chapter Test Solutions

1. A travel agent is trying to schedule a client's trip from city Ato city B. There are three direct flights, three flights from A toa connecting city C, and four flights from this connecting cityC to city B. How many trips are possible? 2. A club has 45 members, and its membership committee hasthree members. How many different membership commi

14,19ttees

are possible?

3. Find the coeffic

82 6of in the expansion of 2 .

4. List the elements in the sample space. A game spinner on a circular region divided into 6 equal sectors numbered 1 - 6 is spun

1,2,3,.

x y x y

5. A fair coin is tossed four times. Find the probability of obtainingone head and three tails. 6. Two cans of mixed nuts of different brands are open on a table.Brand A consists of 30% cashe

ws, while brand B consists of 40%cashews. A can is chosen at random, and a nut is chosen at randomfrom the can. Find the probability that the nut is(a) from the brand A can. (b) a brand A cashew.

0.5 0.1

(c) a cashew. (d) from the brand A can, given that it is a cashew.

5,10, 20, 40, 87. Find the first

0,160; 10,240 6 terms and the 12th term of the sequence given

5 and 2 , for 2. 8. Find an explicit formula for the th term of the arithmetic sequence

5, 1,3

k kb b b k

,7,...1 1 19. Find the sum of the terms of the geometric sequence 3, 1, , ,3 9 27

10. Determine whether the geometric series 3 0.5 converges. If it does,

121/ 27

conved rgit es; s sum. 3

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