1. 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.

2. 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 1012
I hope they purchased the season pass!

3. 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)

4. Unit 2 Review - Counting (4.6-4.7, 5.1-5.2) MDM 4U
Mr. Lieff

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

6. 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

7. 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) = ! = 720
(10 – 3)!

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

9. 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

10. 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) = ! = 1140
(20-3)! 3!

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

13. 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:

14. 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

15. 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

16. 5.2 Pascal’s Triangle
Determine the number of paths using Pascal’s Triangle on a:
Grid
Checkerboard
Word grid
Examples in SMART Notebook

17. 5.2 Pascal’s Triangle

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