Warm up A Ferris wheel holds 12 riders. If there are 20 people waiting in line, how many different ways can 12 people ride it? You may write your answer.

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Presentation transcript:

Warm up A Ferris wheel holds 12 riders. If there are 20 people waiting in line, how many different ways can 12 people ride it? You may write your answer in terms of factorials/permutations/combinations.

Solution Since only 12 of the 20 people can ride the ferris wheel at a time, there are C(20,12) or different groups of riders. Each group can be placed on the ferris wheel 11! or ways since it is a circular permutation. So the total number of ways is: C(20, 12) x 11! = x = or 5.03 x I hope they purchased the season pass!

Unit 2 Review ( , 5.1) MDM 4U Mr. Lieff

Combinatorics (§4.6 & 4.7) Permutations – order matters e.g., President Combinations – order does not matter e.g., Committee

4.6 Permutations Find the number of outcomes given a situation where order matters Calculate the probability of an outcome or outcomes in situations where order matters Recognizing how to restrict the calculations when some elements are the same

4.6 Permutations Ex: How many ways can 5 students be arranged in a line? Ans: 5! = 120 Ex: How many ways are there if Jake must be first? Ans: (5-1)! = 4! = 60 Ex: in a class of 10 people, a teacher must pick 3 for an experiment (students are tested in a particular order) How many ways are there to do this? Ans: P(10,3) = 10! = 720 (10 – 3)!

Permutations cont’d How many ways are there to rearrange the letters in the word TOOLTIME? 8! = (2!2!)

4.6 Permutations Ex: What is the probability of opening one of the school combination locks by chance? First digit cannot be repeated Ans: 1 in 60 x 59 x 59 = 1 in Circular Permutations: There are (n-1)! ways to arrange n objects in a circle

4.7 Combinations Find the number of outcomes given a situation where order does not matter Calculate the probability of an outcome or outcomes in situations where order does not matter Ex: How many ways are there to choose a 3 person committee from a class of 20? Ans: C(20,3) = 20!___ = 1140 (20-3)! 3!

4.7 Combinations Ex: From a group of 5 men and 4 women, how many committees of 5 can be formed with a. exactly 3 women b. at least 3 women ans a: ans b:

5.1 Probability Distributions and Expected Value determine the probability distributions for discrete random variables determine the expected value of a discrete random variable ex: what is the probability distribution for results of rolling an 8 sided die? Roll Prob.⅛⅛⅛⅛⅛⅛⅛⅛

5.1 Probability Distributions and Expected Value ex: what is the expected value for rolling an 8 sided die? ans: E(X) = 1(⅛) + 2(⅛) + 3(⅛) + 4(⅛) + 5(⅛) + 6(⅛) + 7(⅛) + 8(⅛) = 4.5

Review Read your notes! pp #1, 5df, 6, 8, 11; p. 270 #4, 6 pp #1, 2; p. 326 #1