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

4.6 Counting Techniques and Probability Strategies - Permutations How many ways can you arrange a deck of cards? https://www.youtube.com/watch?v=uNS1QvDzCVw Learning goal: Count arrangements of objects when order matters MSIP / Home Learning: pp. 255-257 #1-7, 11, 13, 14, 16

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

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

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

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?

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)

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 = 3 628 800 ways

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!?

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 3 3 In general, there are (n-1)! ways to arrange n unique objects in a circle.

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

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

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?

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?

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

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?

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

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

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

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

Permutations of SOME objects (cont’d)  

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

Picking 3 people from 10… We get 720 possible arrangements

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

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

The Answer so we have 1 chance in 120 or 0.0083 probability

Permutations Summary  

MSIP / Home Learning pp. 255-257 #1-7, 11, 13, 14, 16

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?

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

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

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!

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?

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

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 6 = 0.0045 or 0.45%. 1320 What is the probability they play together on ANY of the 4 lines? 4 x 0.45% = 1.8%

4.7 Counting Techniques and Probability Strategies - Combinations Chapter 4 – Introduction to Probability Learning goal: Count arrangements when order doesn’t matter Questions? pp. 255-257 #1-7, 11, 13, 14, 16 MSIP / Home Learning: pp. 262-265 # 1-3, 5, 7, 9, 18

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

Combinations A combination of 5 students from 20 is calculated the following way, giving 15504 ways for Mr. Lieff to choose his committee.

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

How many combos are possible?

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 = 6 + 4 = 10 Possible combinations = 10 x 2 x 2 x 1 = 40

Yo quiero Taco Bell!

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

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

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?

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 = 36 864 Whataburger also has 2 oz. Jr patties, grilled chicken and breaded Whatachicken patties How does this affect the number of combinations?

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

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

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

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) = 4 = 4 = 1 _ C(52,5) 2598960 649 740 You will likely need to play a lot of poker to get one of these hands!

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

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!

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

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)

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

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

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

References Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page