1. Week 9 - Friday

2.  What did we talk about last time?  Partial orders  Total orders  Basic probability  Event  Sample space  Monty Hall  Multiplication rule

4.  There are 10 thieves who have just stolen an enormous pile of loot: gold, jewels, solid state drives, and so on  The thieves need to find a way to divide it all equally  Give an algorithm such that each of the 10 thieves believe that he is getting at least 1/10 of the loot  Hint: When you were a kid, how did your mother have you and your brother or sister divide the last piece of cake?

6.  A permutation of a set of objects is an ordering of the objects in a row  Consider set { a, b, c }  Its permutations are:  abc  acb  cba  bac  bca  cab  If a set has n  1 elements, it has n! permutations

7.  How many different ways can the letters in the word "WOMBAT" be permuted?  How many different ways can "WOMBAT" be permuted such that "BA" remains together?  What is the probability that, given a random permutation of "WOMBAT", the "BA" is together?  How many different ways can the letters in "MISSISSIPPI" be permuted?  How many would it be if we don't distinguish between copies of letters?

8.  What if you want to seat 6 people around a circular table?  If you only care about who sits next to whom (rather than who is actually in Seat 1, Seat 2, etc.) how many circular permutations are there?  What about for n people?

9.  An r-permutation of a set of n element is an ordered selection of r elements from the set  Example: A 2-permutation of {a, b, c} includes:  ab  ac  ba  bc  ca  cb  The number of r-permutations of a set of n elements is P(n,r) = n!/(n – r)!

10.  What is P(5,2)?  How many 4-permutations are there in a set of 7 objects?  How many different ways can three of the letters in "BYTES" be written in a row?

12.  If a finite set A equals the union of k distinct mutually disjoint subsets A 1, A 2, … A k, then: N(A) = N(A 1 ) + N(A 2 ) + … + N(A k )

13.  How many passwords are there with length 3 or smaller?  Assume that a password is only made up of lower case letters  Passwords with length 3 or smaller fall into 3 disjoint sets  Number of passwords with length 1  Number of passwords with length 2  Number of passwords with length 3  Total passwords = 26 + 26 2 + 26 3 = 18278

14.  If A is a finite set and B is a subset of A, then N(A – B) = N(A) – N(B)  Example:  Recall that a PIN has 4 digits, each of which is one of the 26 letters or one of the 10 digits  How many PINs contain repeated symbols?  What is the probability that a PIN contains a repeated symbol?

15.  If A, B, C are any finite sets, then N(A  B) = N(A) + N(B) – N(A  B)  And, N(A  B  C) = N(A) + N(B) + N(C) – N(A  B) – N(A  C) – N(B  C) + N(A  B  C)

16.  How many integers from 1 through 1,000 are multiples of 3 or multiples of 5?  How many integers from 1 through 1,000 are neither multiples of 3 nor multiples of 5?

17.  Consider a survey of 50 students about the programming languages they know  The results are:  30 know Java  18 know C++  26 know ML  9 known both Java and C++  16 know both Java and ML  8 know both C++ and ML  47 know at least one of the three  How many students know none of the three?  How many students know all three?  How many students know Java and C++ but not ML?  How many students know Java but neither C++ nor ML?

18. Student Lecture

20.  How many different subsets of size r can you take out of a set of n items?  Subset of size 3 out of a set of size 5?  Subset of size 4 out of a set of size 5?  Subset of size 5 out of a set of size 5?  Subset of size 1 out of a set of size 5?  This is called an r-combination, written

21.  In r-permutations, the order matters  In r-combinations, the order doesn't  Thus, the number of r-combinations is just the number of r-permutations divided by the possible orderings

22.  How many ways are there to choose 5 people out of a group of 12?  What if two people don't get along? How many 5 person teams can you make from a group of 12 if those two people cannot both be on the team?

23.  How many five-card poker hands contain two pairs?  If a five-card hand is dealt at random from an ordinary deck of cards, what is the probability that the hand contains two pairs?

24.  What if you want to take r things out of a set of n things, but you are allowed to have repetitions?  Think of it as putting r things in n categories  Example: n = 5, r = 4  We could represent this as x||xx|x|  That's an r x's and n – 1 |'s 12345 xxxx

25.  So, we can think of taking an r-combination with repetitions as choosing r items in a string that is r + n – 1 long and marking those as x's  Consequently, the number of r-combinations with repetitions is

26.  Let's say you grab a handful of 10 Starbursts  Original Starbursts come in  Cherry  Lemon  Strawberry  Orange  How many different handfuls are possible?  How many possible handfuls will contain at least 3 cherry?

27.  This is a quick reminder of all the different ways you can count things: Order MattersOrder Doesn't Matter Repetition Allowed nknk Repetition Not Allowed P(n,k)P(n,k)

30.  Binomial theorem  Probability axioms  Expected values

31.  Work on Homework 7  Due Friday before midnight!  Keep reading Chapter 9