CS70: Lecture Review. 2. Stars/Bars. 3. Balls in Bins. 4. Addition Rules. 5. Combinatorial Proofs. 6. Inclusion/Exclusion
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1 CS70: Lecture Review. 2. Stars/Bars. 3. Balls in Bins. 4. Addition Rules. 5. Combinatorial Proofs. 6. Inclusion/Exclusion
2 The rules! First rule: n 1 n 2 n 3. Product Rule. k Samples with replacement from n items: n k. Sample without replacement: n! (n k! Second rule: when order doesn t matter divide..when possible. Sample without replacement and order doesn t matter: ( n k = n! n choose k (n k!k!.
3 Example: visualize. First rule: n 1 n 2 n 3. Product Rule. Second rule: when order doesn t matter divide..when possible card Poker deals: = 52! 49!. First rule. Poker hands:? Hand: Q,K,A. Deals: Q,K,A, Q,A,K, K,A,Q,K,A,Q, A,K,Q, A,Q,K. = First rule again. Second Rule! Total: 52! 49!3! Choose k out of n. n! Ordered set: (n k! What is? k! First rule again. n! = Total: (n k!k! Second rule
4 Example: visualize First rule: n 1 n 2 n 3. Product Rule. Second rule: when order doesn t matter divide..when possible Orderings of ANAGRAM? Ordered Set: 7! First rule. A s are the same! What is? ANAGRAM A 1 NA 2 GRA 3 M, A 2 NA 1 GRA 3 M,... = = 3! First rule! = 7! 3! Second rule!......
5 Splitting up some money... How many ways can Bob and Alice split 5 dollars? For each of 5 dollars pick Bob or Alice(2 5, divide out order??? 5 dollars for Bob and 0 for Alice: one ordered set: (B,B,B,B,B. 4 for Bob and 1 for Alice: 5 ordered sets: (A,B,B,B,B ; (B,A,B,B,B;... Single way to specify, first Alice s dollars, then Bob s. and so on ?? (B,B,B,B,B (A,B,B,B,B (A,A,B,B,B Second rule of counting is no good here!......
6 Splitting 5 dollars.. How many ways can Alice, Bob, and Eve split 5 dollars. Alice gets 3, Bob gets 1, Eve gets 1: (A,A,A,B,E. Separate Alice s dollars from Bob s and then Bob s from Eve s. Five dollars are five stars:. Alice: 2, Bob: 1, Eve: 2. Stars and Bars:. Alice: 0, Bob: 1, Eve: 4. Stars and Bars:. Each split is a sequence of stars and bars. Each sequence of stars and bars is a split. Counting Rule: if there is a one-to-one mapping between two sets they have the same size!
7 Stars and Bars. How many different 5 star and 2 bar diagrams?. 7 positions in which to place the 2 bars. Alice: 0; Bob 1; Eve: 4. Bars in first and third position. Alice: 1; Bob 4; Eve: 0. Bars in second and seventh position. ( 7 2 ways to do so and ( 7 2 ways to split 5 dollars among 3 people.
8 Stars and Bars. Ways to add up n numbers to sum to k? or k from n with replacement where order doesn t matter. In general, k stars n 1 bars.. n + k 1 positions from which to choose n 1 bar positions. ( n + k 1 n 1 Or: k unordered choices from set of n possibilities with replacement. Sample with replacement where order doesn t matter.
9 Summary. First rule: n 1 n 2 n 3. k Samples with replacement from n items: n k. Sample without replacement: n! (n k! Second rule: when order doesn t matter (sometimes can divide... Sample without replacement and order doesn t matter: ( n k = n! n choose k (n k!k!. One-to-one rule: equal in number if one-to-one correspondence. pause Bijection! Sample with replacement and order doesn t matter: ( k+n 1 n 1.
10 Balls in bins. k Balls in n bins k samples from n possibilities. indistinguishable balls order doesn t matter only one ball in each bin without replacement 5 balls into 10 bins 5 samples from 10 possibilities with replacement Example: 5 digit numbers. 5 indistinguishable balls into 52 bins only one ball in each bin 5 samples from 52 possibilities without replacement Example: Poker hands. 5 indistinguishable balls into 3 bins 5 samples from 3 possibilities with replacement and no order Dividing 5 dollars among Alice, Bob and Eve.
11 Sum Rule Two indistinguishable jokers in 54 card deck. How many 5 card poker hands? Sum rule: Can sum over disjoint sets. No jokers exclusive or One Joker exclusive or Two Jokers ( 52 ( ( Two distinguishable jokers in 54 card deck. How many 5 card poker hands? Choose 4 cards plus one of 2 jokers! ( ( 52 4 ( Wait a minute! Same as choosing 5 cards from 54 or ( 54 5 Theorem: ( 54 ( 5 = 52 ( ( Algebraic Proof: Why? Just why? Especially on Friday! Above is combinatorial proof.
12 Combinatorial Proofs. Theorem: ( n ( k = n n k Proof: How many subsets of size k? ( n k How many subsets of size k? Choose a subset of size n k and what s left out is a subset of size k. Choosing a subset of size k is same as choosing n k elements to not take. = ( n n k subsets of size k.
13 Pascal s Triangle Row n: coefficients of (1 + x n = (1 + x(1 + x (1 + x. Foil (4 terms on steroids: 2 n terms: choose 1 or x froom each factor of (1 + x. Simplify: collect all terms corresponding to x k. Coefficient of x k is ( n k : choose k factors where x is in product. ( 0 ( 0 1 ( ( 2 ( 2 ( ( 3 ( 3 ( 3 ( Pascal s rule = ( n+1 ( k = n ( k + n k 1.
14 Combinatorial Proofs. Theorem: ( n+1 ( k = n ( k + n k 1. Proof: How many size k subsets of n + 1? ( n+1 k. How many size k subsets of n + 1? How many contain the first element? Chose first element, need to choose k 1 more from remaining n elements. = ( n k 1 How many don t contain the first element? Need to choose k elements from remaining n elts. = ( n k ( = n+1 k. So, ( n k 1 + ( n k
15 Combinatorial Proof. Theorem: ( n ( k = n 1 ( k k 1 k 1. Proof: Consider size k subset where i is the first element chosen. {1,...,i,...,n} Must choose k 1 elements from n i remaining elements. = ( n i k 1 such subsets. Add them up to get the total number of subsets of size k which is also ( n+1 k.
16 Binomial Theorem: x = 1 Theorem: 2 n = ( n ( n + n ( n n 0 Proof: How many subsets of {1,...,n}? Construct a subset with sequence of n choices: element i is in or is not in the subset: 2 poss. First rule of counting: = 2 n subsets. How many subsets of {1,...,n}? ( n i ways to choose i elts of {1,...,n}. Sum over i to get total number of subsets..which is also 2 n.
17 Simple Inclusion/Exclusion Sum Rule: For disjoint sets S and T, S T = S + T Used to reason about all subsets by adding number of subsets of size 1, 2, 3,... Also reasoned about subsets that contained or didn t contain an element. (E.g., first element, first i elements. Inclusion/Exclusion Rule: For any S and T, S T = S + T S T. Example: How many 10-digit phone numbers have 7 as their first or second digit? S = phone numbers with 7 as first digit. S = 10 9 T = phone numbers with 7 as second digit. T = S T = phone numbers with 7 as first and second digit. S T = Answer: S + T S T =
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