ProbabilityProbability

53 Fundamental counting principle52 Factorials51 Permutations50 WP: Permutations 49 Combinations48 WP: Combinations

QuoteQuote

It is easier to gain It is easier to gain forgiveness than to forgiveness than to

get permission.get permission.

Grace Murray HopperGrace Murray Hopper

Puzzle – What is this?Puzzle – What is this?

The maker doesn’t want The maker doesn’t want it, the buyer doesn’t use it, the buyer doesn’t use it and the user doesn’t it and the user doesn’t see it. see it.

What is it?What is it?

53a Fundamental counting principle53a Fundamental counting principle

Fundamental Counting Principal =Fundamental Counting Principal = Fancy Fancy way of describing how one would way of describing how one would determine the number of ways a sequence determine the number of ways a sequence of events can take place. of events can take place.

53b Fundamental counting principle53b Fundamental counting principleYou are at your school cafeteria that allows you to choose a You are at your school cafeteria that allows you to choose a lunch meal from a set menu. You have lunch meal from a set menu. You have twotwo choices for the choices for the Main course (a hamburger or a pizza), Main course (a hamburger or a pizza), TwoTwo choices of a drink choices of a drink (orange juice, apple juice) and (orange juice, apple juice) and ThreeThree choices of dessert (pie, choices of dessert (pie, ice cream, jello). ice cream, jello).

How many different meal combos can you select?_________ How many different meal combos can you select?_________

Method one: Tree diagramMethod one: Tree diagram LunchLunch

HamburgerHamburger PizzaPizza

AppleApple OrangeOrange Apple Orange

PiePieIcecreamIcecream

JelloJello

PiePieIcecreamIcecream

JelloJello

PiePieIcecreamIcecream

JelloJello

PiePieIcecreamIcecream

JelloJello

12 meals12 meals

53c Fundamental counting principle53c Fundamental counting principle

Method two:Method two: Multiply number of choicesMultiply number of choices

2 x 2 x 3 = 2 x 2 x 3 = 12 meals12 meals

Ex 2: No repetitionEx 2: No repetition

During the Olympic 400m sprint, there are 6 runners. How During the Olympic 400m sprint, there are 6 runners. How many possible ways are there to award first, second, and many possible ways are there to award first, second, and third places?third places?

3 places3 places ____ x ____ x ____ =____ x ____ x ____ =66 55 44 120 different ways120 different ways1st1st 2nd2nd 3rd3rd

53d Fundamental counting principle53d Fundamental counting principleEx 3: With repetitionLicense Plates for cars are labeled with 3 letters followed by 3 digits. (In this case, digits refer to digits 0 - 9. If a question asks for numbers, its 1 - 9 because 0 isn't really a number)How many possible plates are there? You can use the same number more than once.

___ x ___ x ___ x ___ x ___ x ___ =26 26 26 10 10 10 17,576,000 plates

Ex 4: Account numbers for Century Oil Company consist of five digits. If the first digit cannot be a 0 or 1, how many account numbers are possible?

___ x ___ x ___ x ___ x ___ =8 10 10 10 10 80,000 different account #’s

53e Fundamental counting principle53e Fundamental counting principleWe are going to collect data from cars in the student parking lot.

License place Vehicle color

1234.....50

Factorials - QuoteFactorials - Quote

Space and time are Space and time are intimately intertwined and intimately intertwined and indissolubly connected indissolubly connected with each other.with each other.

Sir William Rowan HamiltonSir William Rowan Hamilton

Factorials - PuzzleFactorials - Puzzle

There is a square fountain that has a tree There is a square fountain that has a tree growing at each corner. I want to turn this growing at each corner. I want to turn this into a piranha pond, but to do that the size into a piranha pond, but to do that the size of the fountain needs to be doubled. How of the fountain needs to be doubled. How could I do this without digging deeper or could I do this without digging deeper or moving a tree and still have a square moving a tree and still have a square fountain?fountain?

52a Factorials52a Factorials

5 • 4 • 3 • 2 • 1 = 5! Factorial

7!= 7 • 6 • 5 • 4 • 3 • 2 •1 = 5040

!5

!7

12345

123456742

!3!5

!8

12312345

12345678

123

678

1

7856

QuoteQuote

Algebra is but written Algebra is but written geometry, and geometry, and

geometry is but written geometry is but written algebra.algebra.

Sophie GermainSophie Germain

PuzzlePuzzleWhat are the last What are the last few hairs on a few hairs on a dogs tail called?dogs tail called?

51a Permutations 51a Permutations Permutations = A listing in which order IS important.

Can be written as: P(6,4) or 6P4

P(6,4) Represents the number of ways 6 items can be taken 4 at a time…..

Or 6 x 5 x 4 x 3 = 360

Find P(15,3) = _____2730

Or 6 (6-1) (6-2) (6-3)

15 x 14 x 13

51b Permutations - Activity51b Permutations - Activity

Write the letters G R A P H on the top of your paper.

Compose a numbered list of different 5 letter Permutations. -(not necessarily words)

On the bottom of your paper write how many different permutations you have come up with.

Don’t forget your Name, Date and Period before turning in.

Hint: You may wish to devise a strategy or pattern for finding all of the permutations before you start.

QuoteQuote

Happy is the man who devotes Happy is the man who devotes himself to a study of the himself to a study of the heavens ... their study will heavens ... their study will furnish him with the pursuit of furnish him with the pursuit of enjoyments.enjoyments.

Johannes KeplerJohannes Kepler

PuzzlePuzzleFrom statistical records, what is the most dangerous job in America?

50a WP: Permutations50a WP: PermutationsUse the same formula from section 52 to solve these WPs.

Ex1. Ten people are entered in a race. If there are no ties, in how many ways can the first three places come out?

___ x ___ x ___ =10 9 8 720

Ex2. How many different arrangements can be made with

the letters in the word LUNCH?5! or ___ x ___ x ___ x ___ x ___ =5 4 3 2 1 120

Ex3. You and 8 friends go to a concert. How many different ways can you sit in the assigned seats?

9! = 362,880

50b WP: Permutations - Activity 50b WP: Permutations - Activity

On a separate sheet of paper, use only the letters below to form as many words as possible. Don’t forget Name, Date and Period.Don’t forget Name, Date and Period.

Mathematics PermutationsMathematics Permutations1234.....50

QuoteQuote

In mathematics there are In mathematics there are no true controversies.no true controversies.

Karl Friedrich Gauss (gowse)Karl Friedrich Gauss (gowse)

PuzzlePuzzle

How long will a so-How long will a so-called Eight Day called Eight Day Clock run without Clock run without winding?winding?

49a Combinations49a Combinations

Combinations = A listing in which order is NOT important.

Can be written as: C(3,2) or 3C2

C(3,2) means the number of ways 3 items can betaken 2 at a time. (order does not matter)

Ex. C(3,2) using the letters C A T

CA   CT   AT

                          

!r

PC rn

rn n = totalr = What you want

49b Combinations49b Combinations

!r

PC rn

rn n = totalr = What you want

C(7,2) !227

27

PC 7 x 6

2 x 1=

422

= 21

Which is not an expression for the number of ways 3 items can be selected from 5 items when order is not considered?

QuoteQuote

Say what you know, Say what you know, do what you must, do what you must,

come what may.come what may.Sonya Kovalevsky (co va LEV ski)Sonya Kovalevsky (co va LEV ski)

WP: CombinationsWP: Combinations

If you were to take two apples If you were to take two apples from three apples, how many from three apples, how many

would you have?would you have?

48a WP: Combinations48a WP: Combinations

Permutations = Order IS important

P(8,3) = ___ x ___ x ___8 7 6 = 336

Combinations = Order does not matter

C(8,3) = !3

)3,8(P

23

336

6

33656

48b WP: Combinations48b WP: Combinations

Ex1. A college has seven instructors qualified to teach a special computer lab course which requires two instructors to be present. How many different pairs of teachers could there be?

C(7,2) = !2

6721

Ex2. A panel of judges is to consist of six women and three men. A list of potential judges includes seven women and six men. How many different panels could be created from this list?

Women MenC(7,6)

123456

2345677

C(6,3) =

123

456

7*20 = 140 140 choices

20