Warm up (TIPS) 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 125 970 different groups of riders. Each group can be placed on the ferris wheel 11! or 39 916 800 ways since it is a circular permutation. So the total number of ways is: C(20, 12) x 11! = 125 970 x 39 916 800 = 5 028 319 296 000 or 5.03 x 1012 I hope they purchased the season pass!

U2 Test WedSenior Privileges Thu U2 Test Games Fair is due by 2:20 Similar in length to U1 Test 15 MC, 3 KU, 4 APP, 2 TIPS 1 hour limit (unless IEP)

Unit 2 Review - Counting (4.6-4.7, 5.1-5.2) 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 Recognize 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 Rod and Todd must be next to each other? Ans: 4! x 2! = 48 Count Rod and Todd as one person, but in every arrangement they can be arranged 2! ways Ex: in a class of 10 people, a teacher must pick 3 for an experiment to be tested in 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! = 10 080 (2!2!)

4.6 Permutations Ex: What is the probability of opening one of the school combination locks (0-39) by chance if the second digit must be unique? Ans: 1 in 40 x 39 x 39 = 1 in 60840 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!

Coca-Cola Freestyle Standard is 20 flavours and 2 choices Either number could change for a ‘different’ question

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 Dice, coins Determine the expected value of a discrete random variable Ex: what is the probability distribution for results of rolling an 8 sided die? Roll 1 2 3 4 5 6 7 8 Prob. ⅛

5.1 Probability Distributions and Expected Value Ex: what is the expected value for rolling an 8 sided die? Multiply every outcome by its probability and sum the products Ans: E(X) = 1(⅛) + 2(⅛) + 3(⅛) + 4(⅛) + 5(⅛) + 6(⅛) + 7(⅛) + 8(⅛) = 4.5

5.2 Pascal’s Triangle Determine the number of paths using Pascal’s Triangle on a: Grid Checkerboard Word grid Examples in SMART Notebook

5.2 Pascal’s Triangle  

Review Read your notes! Complete the Games Fair! pp. 268-269 #1, 5df, 6, 8, 11; p. 270 #3, 4, 6 pp. 324 #1-6; p. 326 #1, 3, 6-7