Transcript Unit 2 Review

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
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!
Unit 2 Review - Counting
(4.4?, 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 by chance?

First digit cannot be repeated
 Ans: 1 in 60 x 59 x 59 = 1 in 208 860
 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  4 
    10  4  40
 2  3 
 5  4   5  4 
        45
 2  3   1  4 
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
1
2
3
4
5
6
7
8
Prob.
⅛
⅛
⅛
⅛
⅛
⅛
⅛
⅛
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
5.2 Pascal’s Triangle
 Determine the number of paths using
Pascal’s Triangle



Grid
Checkerboard
Word grid
 Ex: in SMART Notebook
5.2 Pascal’s Triangle and the Binomial
Theorem

Review
 Complete all Home Learning!
 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, 6