Warm Up 1.A restaurant offers a Sunday brunch. With your meal you have your choice of 3 salads, 4 sides, 3 entrees and 5 beverages and you can have either one of 4 ice creams or 6 pies. How many configurations can be made? 2. Convert.087 to a percent. 3. A bag contains the following lettered tiles: s,t,a,t,I,s,t,I,c,s –Probability of picking a vowel? ________ –Probability of picking c or s? __________

Warm Up 1. There are six point in a plane, but no three can be collinear. How many different straight lines that pass through two of the points are possible? 2. Eric has 3 sandwiches, 2 salads, and 4 drinks. He will choose a sandwich, a salad and a drink OR he will choose a salad and a drink. Find the number of combinations possible. 3. Maryann has 6 red apples, 4 yellow apples and 2 green apples in a bucket. What is the probability that Maryann will choose a red or green apple?

Unit 4 The Chances of Winning MM1D1: Students will determine the number of outcomes related to a given event

FACTORIALS The notation n ! is called n factorial, and n ! = n (n - 1) (n - 2) X... X 3 X 2 X 1 By definition, 0! = 1.

Examples 5! = 5 X 4 X 3 X 2 X 1=120 6! = 3! = 9! =

Factorials If it is a division problem you need to expand and then cancel any integers they have in common before multiplying.

Factorials

PERMUTATIONS Permutation is just another word for an ordered arrangement or an ordered sequence of distinct objects (repetition is not allowed). You can think of the word “permutation” as an “arrangement.”

Definition Any ordered sequence of r objects taken from a set of n distinct objects is called a permutation.

Example A contractor builds homes of 8 different models and has 5 lots to build on. In how many different ways can he place the model homes on these lots? That is, we wish to count the number of arrangements of 8 homes taken 5 at a time.

Example A library has 9 textbooks, labeled # 1-9, but only 5 spots on the shelf. In how many different ways can we place the books on the shelf?

Check Up 1. 7! Drake can choose from 31 flavors of ice cream. He wants to get a bowl with four scoops of ice cream. Each of the four scoops of ice cream will be a different flavor. How many different bowls are possible. 4. Danny can choose from 31 flavors of ice cream and 3 types of cones. He can then either have 5 different toppings or 3 different sauces. How many combinations are possible?

COMBINATIONS The ways of selecting a subset of a set, where the order in which the elements are selected is of no importance, are called combinations. (Repetition is still not allowed.)

The number of combinations (or subsets) of n distinct things taken r at a time is given by

Example You need to send a 3- person group from a class of 5 students to represent the class at a contest. How many different groups are there?

Example You need to send a 4-person group from a class of 5 students to represent the class at a contest. How many different groups are there?

Example How many 5-card hands can be dealt from a 52-card deck of cards?

Check Up 1. There are 14 students in a mathematics competition. Each student will earn points during the competition. There are prizes for the top three finishers. How many different ways can the students finish? 2. There are 14 students in a mathematics competition. Each Student will earn points during the competition. There are prizes for the top three finishers. How many different arrangements can the students finish in?

QUIZ TOMORROW! Your quiz tomorrow will be covering: –Factorials –Permutations –Combinations –Both Counting Principles –BE PREPARED!!!!!!!!