1. Minds on!
If you shuffled a standard deck of cards once per second, how long do you think it would take before every possible order appeared?
A) Before 2018
B) Before 2027
C) Before 2100
D) None of the above

2. 4.6 Counting Techniques and Probability Strategies - Permutations
How many ways can you arrange a deck of cards?
Learning goal: Count arrangements of objects when order matters
MSIP / Home Learning: pp #1-7, 11, 13, 14, 16

3. Overall Expectations – Counting and Probability Distributions (4 lessons)
A2. Solve problems involving the counting of ordered and unordered objects to determine the probability of an event
B1. Demonstrate an understanding of discrete probability distributions, represent them numerically, graphically, and algebraically, determine expected values, and solve related problems from a variety of applications

4. Arranging blocks when order matters
Groups A-2: 1a Arranging unique blocks in a line
Collect 6 blocks of different colours
Groups 3-4: 1b Arranging unique blocks in a circle
Groups 5-7: 1c Arranging blocks in a line when some are identical
Collect 3 blocks of one colour and 3 of different colours
Exploration time: 20 min

5. Jigsaw Share your findings with your new group Clubs rotate CW 1 group
Diamonds rotate CCW 1 group
Hearts hold
Share your findings with your new group
Concentrate on the general case for n blocks

6. Activity 1a – Arranging unique blocks in a line
Start with 3 cube-a-links of different colours
How many ways can you arrange them in a line on your desk?
Record the number in the first column.
How about 4 blocks? 5? 6?
What is the pattern?

7. Selecting When Order Matters
There are fewer choices for later places
For 3 blocks:
First block - 3 choices
Second block - 2 choices left
Third block - 1 choice left
Number of arrangements for 3 blocks is
3 x 2 x 1 = 6
There is a mathematical notation for this (and your calculator has it)

8. Factorial Notation (n!)
n! is read “n-factorial”
n! = n x (n – 1) x (n – 2) x … x 2 x 1
n! is the number of ways of arranging n unique objects when order matters
e.g.,
3! = 3 x 2 x 1 = 6
5! = 5 x 4 x 3 x 2 x 1 = 120
NOTE: 0! = 1
Ex. If we have 10 books to place on a shelf, how many possible ways are there to arrange them?
10! = 10 x 9 x 8 x … x 2 x 1 = ways

9. Activity 1b – Arranging Unique Blocks in a Circle
Start with 3 blocks of different colours
Arrange them in a circle
Find the number of different arrangements / orders
Repeat for 4
Try it for 5
Do you see a pattern?
How does it connect to n!?

10. Circular Permutations
For 3 blocks there are 6 different arrangements
ABC, BCA, CAB
ACB, CBA, BAC
However, there are only 2 different orders
Every other arrangement is a rotation of one of these
So the number of DIFFERENT arrangements is
3! = 3x2x1 = 2x1 = 2
In general, there are (n-1)! ways to arrange n unique objects in a circle.

11. Circular Permutations
How many arrangements are there of 6 old chaps around a table?

12. Circular Permutations
There are 6! ways to arrange the 6 old chaps
However, if everyone shifts one seat, the arrangement is the same
This can be repeated 4 more times (6 total)
Therefore 6 of each arrangement are identical
So the number of DIFFERENT arrangements is
6! = 6x5x4x3x2x1 = 5x4x3x2x1 = 5! = 120
6 6

13. Activity 1c – Arranging blocks in a line when some are identical
Start with 3 blocks where 2 are the same colour
How many ways can you arrange them in a line?
How does this number compare to the number of ways of arranging 3 unique blocks?
Repeat for 4 blocks where 2 are the same
Try it for 5 where 2 are the same
Do you see a pattern?

14. Permutations When Some Objects Are Alike
Suppose you are creating arrangements and some objects are alike
For example, the word ear has 3! or 6 arrangements (aer, are, ear, era, rea, rae)
But the word eel has repeating letters and only 3 arrangements (eel, ele, lee)
How do we calculate arrangements in these cases?

15. Permutations When Some Objects Are Alike
To perform this calculation we divide the number of possible arrangements by the arrangements of objects that are identical
n is the number of objects
a, b, c are the number of objects that occur more than once

16. So back to our problem Arrangements of the letters in the word eel
What is the number of arrangements of 8 socks if 3 are red, 2 are blue, 1 is black, one is white and one is green?

17. Another Example
How many arrangements are there of the letters in the word BOOKKEEPER?

18. Warm up
How many arrangements are there of the letters in the word MISSISSIPPI?

19. Recap n! is read “n-factorial”
n! = n(n-1)(n-2)…(2)(1) is the number of ways of arranging n unique objects in a row
n! = (n-1)! is the number of ways of arranging n n unique objects in a circle
n!__ is the number of ways of arranging n
a!b!c!... objects when a, b, c, … are identical

20. Permutations of SOME objects
Suppose we have a group of 10 people. How many ways are there to pick a president, vice-president and treasurer?
In this case we are selecting people for a particular order

21. Permutations of SOME objects (cont’d)

22. Permutation Notation
A permutation is an ordered arrangement of objects selected from a set
Written P(n,r) or nPr
The number of possible arrangements of r objects from a set of n objects

23. Picking 3 people from 10…
We get 720 possible arrangements

24. Arrangements With Replacement
Suppose you were looking at arrangements where you replaced the object after you had chosen it
If you draw two cards from the deck, you have 52 x 51 possible arrangements
If you draw a card, replace it and then draw another card, you have 52 x 52 possible arrangements
Replacement increases the possible arrangements

25. Permutations and Probability
10 different coloured socks in a drawer
What is the probability of picking green, red and blue in any order?

26. The Answer
so we have 1 chance in 120 or probability

27. Permutations Summary

28. MSIP / Home Learning
pp #1-7, 11, 13, 14, 16

29. Warm up
The Coca-Cola freestyle machine boasts “100+ drink choices” (2 flavours)
How many different flavour combinations are possible if there are 20 flavours?

30. A proposed solution… Number of permutations of 2 flavours
= 380
But…does order matter?
Is lime/Coke different from Coke/lime?
No…so we counted every dual flavor twice!
There are actually 380÷2 = 190 drink combinations
This is because there are 2! = 2 arrangements of every pair of flavours

31. Warm up
i) How many ways can 8 children be placed on an 8-horse Merry-Go-Round?
ii) What if Simone and Rachel wanted to sit on adjacent horses?
i) 7! = 5 040
ii) Group Simone and Rachel. Thus, we are only arranging 7 children on 7 horses, so 6! = 720
However, there are 2! ways to arrange Simone and Rachel on their horses, so 720x2 = 1440

32. Warm up There are 20 different prizes in Mr. Lieff’s prize box.
How many ways can Sal, Val and Hal choose prizes?
Here, order matters, since there are 3 different prizes for 3 different people
P(20, 3) = 20! = 20! = 20x19x18 = 6 840
(20-3)! 17!

33. Warm up
At the 2013 NHL All Star Game, Team Alfredsson featured 3 Ottawa Senators forwards. If head coach John Tortorella randomly selected his lines from a group of 12 forwards, what is the probability that the three Senators played together on the first line? On any line?

34. Drawing a diagram On the first line, there are:LW C RW
Line 1 3 2 1
Line 2 X
Line 3
Line 4
LW C RW
On the first line, there are:
3 choices for the first position
2 choices for the second position
1 choice for the third position

35. Solution
There are 3! = 6 different ways to slot the 3 Sens on the first line.
There are P(12, 3) = (12)! = 1320 possible line
(12-3)! line combinations.
So the probability is = or 0.45%.
1320
What is the probability they play together on ANY of the 4 lines?
4 x 0.45% = 1.8%

36. 4.7 Counting Techniques and Probability Strategies - Combinations
Chapter 4 – Introduction to Probability
Learning goal: Count arrangements when order doesn’t matter
Questions? pp #1-7, 11, 13, 14, 16
MSIP / Home Learning: pp # 1-3, 5, 7, 9, 18

37. When Order is Not Important
A combination is an unordered selection of elements from a set
There are many times when order is not important
Suppose Mr. Lieff has 20 MDM4U students and must choose a Data Fair Committee of 5
Order is not important
We use the notation C(n,r) or nCr where n is the number of elements in the set and r is the number we are choosing

38. Combinations
A combination of 5 students from 20 is calculated the following way, giving ways for Mr. Lieff to choose his committee.

39. An Example of a Restriction on a Combination
Suppose that one of Mr. Lieff’s students is the principal’s daughter
It would be a good career move to make her one of the 5 committee members
Here there are really only 4 choices from 19 students
So the calculation is
C(19,4) = 3876
Now there are only 3876 possible combinations for the committee

40. How many combos are possible?

41. Combinations from Complex Sets
Combinations of 2 different mains = C(4,2) = 6
Combinations of sides = C(2,1) = 2
Combinations of desserts = C(2,1) = 2
Combinations of drinks = C(1,1) = 1
Possible combinations =
C(4,2) x C(2,1) x C(2,1) x C(1,1) = 6 x 2 x 2 x 1 = 24
You have 24 possible meals, so you better make a run for the border!
Uno momento por favor…
What if you order 2 of the same main?
There are 4 ways to do this
So Combinations of mains = = 10
Possible combinations = 10 x 2 x 2 x 1 = 40

42. Yo quiero Taco Bell!

43. Whataburger?
What combinations need to be considered for a burger?

44. How many different ice cream flavours are possible
How many different ice cream flavours are possible? (53 toppings, 4 sauces)

45. Minds on…Whataburger! Meat options: 4 oz. patty single, double, triple
Bread options: 5” white or wheat bun, Texas toast, 4” white bun
Plain, toasted on grill, toasted on both sides
Condiments: mayonnaise, mustard, Fancy Ketchup
Veggies: lettuce, tomatoes, pickles, onions
Extras: cheese, jalapenos, bacon
How many different Whataburgers are possible?

46. Whataburger! Combinations
Meat options: C(3, 1) = 3
Bread options: C(4, 1) x 3 = 4 x 3 = 12
Condiments: 2x2x2 = 8
Veggies: 2x2x2x2 = 16
Extras: 2x2x2 = 8
Total: 3x12x8x16x8 =
Whataburger also has 2 oz. Jr patties, grilled chicken and breaded Whatachicken patties
How does this affect the number of combinations?

47. Calculating the Number of Combinations
Suppose you are playing coed volleyball, with a team of 4 men and 5 women
The rules state that you must have at least 3 women on the floor at all times (6 players)
How many combinations of team lineups are there?
You need to take into account team combinations with 3, 4, or 5 women

48. Solution 1: Direct Reasoning
In direct reasoning, you determine the number of possible combinations of suitable outcomes and add them
Find the combinations that have 3, 4 and 5 women and add them

49. Solution 2: Indirect Reasoning
In indirect reasoning, you determine the total possible combinations of outcomes and subtract unsuitable combinations
Find the total combinations and subtract those with 2 women

50. Finding Probabilities Using Combinations
What is the probability of drawing a Royal Flush (10-J-Q-K-A of the same suit) from a deck of cards?
There are C(52,5) ways to draw 5 cards
There are 4 ways to draw a royal flush
P(Royal Flush) = = = _ C(52,5)
You will likely need to play a lot of poker to get one of these hands!

51. Finding Probability Using Combinations
What is the probability of drawing 4 of a kind?
Choose 1 of 13 cards for the 4-of-a-kind
Choose all 4 of those cards
Choose 1 of the remaining 48 cards

52. Extra Challenge Pick a few different hands and find the probabilities
What is more likely: hand X or hand Y? would make a good TIPS test question!

53. World Series Odds 24-Sep-15
Team – win:bet Toronto Blue Jays – 4:1 Kansas City Royals – 11:2 Los Angeles Dodgers – 6:1 New York Mets – 6:1 St. Louis Cardinals – 6:1 Texas Rangers – 9:1 Chicago Cubs – 10:1 Pittsburgh Pirates – 10:1 New York Yankees – 14:1 Houston Astros – 25:1 Los Angeles Angels – 33:1 Minnesota Twins – 33:1

54. Probability and Odds These two terms have different uses in math
Probability involves comparing the number of favorable outcomes with the total number of possible outcomes
If you have 5 green socks and 8 blue socks in a drawer the probability of drawing a green sock is 5/13
Odds compare the number of favorable outcomes with the number of unfavorable outcomes
With 5 green and 8 blue socks, the odds of drawing a green sock is 5 to 8 (or 5:8)

55. World Series Odds 24-Sep-15
Team Odds Probability Toronto Blue Jays 4:1 1/5 Kansas City Royals 11:2 2/13 Los Angeles Dodgers 6:1 1/7 New York Mets 6:1 1/7 St. Louis Cardinals 6:1 1/7 Texas Rangers 9:1 1/10 Chicago Cubs 10:1 1/10 Pittsburgh Pirates 10:1 1/11 New York Yankees 14:1 1/15 Houston Astros 25:1 1/26 Los Angeles Angels 33:1 1/34 Minnesota Twins 33:1 1/34

56. Combinatorics Summary
Permutationsorder matters
e.g., Presidency
Combinationsorder doesn’t matter
e.g., Committee

57. MSIP / Home Learning
pp. 262 – 265 # 1, 2, 3, 5, 7, 9, 18

58. References
Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from