Probability 53 Fundamental counting principle 52 Factorials 51 Permutations 50 WP: Permutations 49 Combinations 48 WP: Combinations 53 Fundamental counting principle 52 Factorials 51 Permutations 50 WP: Permutations 49 Combinations 48 WP: Combinations

Quote It is easier to gain forgiveness than to get permission. Grace Murray Hopper

Puzzle – What is this? The maker doesn’t want it, the buyer doesn’t use it and the user doesn’t see it. What is it?

53a Fundamental counting principle Fundamental Counting Principal = Fancy way of describing how one would determine the number of ways a sequence of events can take place. Fancy way of describing how one would determine the number of ways a sequence of events can take place.

53b Fundamental counting principle You are at your school cafeteria that allows you to choose a lunch meal from a set menu. You have two choices for the Main course (a hamburger or a pizza), Two choices of a drink (orange juice, apple juice) and Three choices of dessert (pie, ice cream, jello). How many different meal combos can you select?_________ Method one: Tree diagram Lunch HamburgerPizza AppleOrange Pie Icecream Jello 12 meals

53c Fundamental counting principle Method two: Multiply number of choices 2 x 2 x 3 = 12 meals Ex 2: No repetition During the Olympic 400m sprint, there are 6 runners. How many possible ways are there to award first, second, and third places? 3 places____ x ____ x ____ = different ways 1st2nd3rd

53d Fundamental counting principle Ex 3: With repetition License Plates for cars are labeled with 3 letters followed by 3 digits. (In this case, digits refer to digits If a question asks for numbers, its 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 ___ = ,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 ___ =810 80,000 different account #’s

53e Fundamental counting principle We are going to collect data from cars in the student parking lot. License placeVehicle color

Factorials - Quote Space and time are intimately intertwined and indissolubly connected with each other. Sir William Rowan Hamilton

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

52a Factorials =5!Factorial 7!= =

Quote Algebra is but written geometry, and geometry is but written algebra. Sophie Germain

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

51a Permutations Permutations =A listing in which order IS important. Can be written as: P(6,4)or 6 P 4 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 - 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.

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

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

50a WP: Permutations Use 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 ___ = Ex2. How many different arrangements can be made with the letters in the word LUNCH? 5! or___ x ___ x ___ x ___ x ___ = 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 Don’t forget Name, Date and Period. 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. Mathematics Permutations

Quote In mathematics there are no true controversies. Karl Friedrich Gauss (gowse)

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

49a Combinations Combinations =A listing in which order is NOT important. Can be written as: C(3,2)or 3 C 2 C(3,2) means the number of ways 3 items can be taken 2 at a time. (order does not matter) Ex. C(3,2) using the letters C A T CA CT AT n = total r = What you want

49b Combinations n = total r = What you want C(7,2) 7 x 6 2 x 1 = 42 2 = 21 Which is not an expression for the number of ways 3 items can be selected from 5 items when order is not considered?

Quote Say what you know, do what you must, come what may. Sonya Kovalevsky (co va LEV ski)

WP: Combinations If you were to take two apples from three apples, how many would you have?

48a WP: Combinations Permutations =Order IS important P(8,3) = ___ x ___ x ___8 7 6 = 336 Combinations = Order does not matter C(8,3) = 56

48b 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) = 21 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? WomenMen C(7,6) 7 C(6,3) = 7*20 = 140  140 choices 20