Announcements CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations I Homework 6 is out today; due next Thursday I Can leave answer in the form 232 – don’t have to calculate exact answer Suppose there is a flock of 36 pigeons and a set of 35 pigeonholes I Each pigeon wants to sit in one hole I But since there are less holes than there are pigeons, one pigeon is left without a hole. Permutations and Combinations 2/37 I Consider an event with 367 people. Is it possible no pair of people have the same birthday? I Consider function f from a set with k + 1 or more elements to a set with k elements. Is it possible f is one-to-one? I Consider n married couples. How many of the 2n people must be selected to guarantee there is at least one married couple? The Pigeonhole Principle: If n + 1 or more objects are placed into n boxes, then at least one box contains 2 or more objects CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig, 3/37 Generalized Pigeonhole Principle I If n objects are placed into k boxes, then there is at least one box containing at least dn/k e objects I Proof: (by contradiction) Suppose every box contains less than dn/k e objects CS311H: Discrete Mathematics Permutations and Combinations 4/37 Examples I I If there are 30 students in a class, at least how many must be born in the same month? I What is the minimum # of students required to ensure at least 6 students receive the same grade (A, B, C, D, F)? I I I I What is the min # of cards that must be chosen to guarantee three have same suit? I I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Examples I Instructor: Işıl Dillig, Homework 5 is due today Instructor: Işıl Dillig, 1/37 The Pigeonhole Principle I I CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig, 5/37 1 CS311H: Discrete Mathematics Permutations and Combinations 6/37 Permutations I I How Many Permutations? A permutation of a set of distinct objects is an ordered arrangement of these objects I No object can be selected more than once I Order of arrangement matters I Consider set S = {a1 , a2 , . . . an } I How many permutations of S are there? I Decompose using product rule: Example: S = {a, b, c}. What are the permutations of S ? I How many ways to choose first element? I How many ways to choose second element? I ... I How many ways to choose last element? I I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 7/37 Instructor: Işıl Dillig, Examples CS311H: Discrete Mathematics Permutations and Combinations 8/37 r -Permutations I Consider the set {7, 10, 23, 4}. How many permutations? I How many permutations of letters A, B, C, D, E, F, G contain ”ABC” as a substring? I I I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 9/37 r-permutation is ordered arrangment of r elements in a set S I S can contain more than r elements I But we want arrangement containing r of the elements in S I The number of r-permutations in a set with n elements is written P (n, r ) I Example: What is P (n, n)? I Instructor: Işıl Dillig, Computing P (n, r ) CS311H: Discrete Mathematics Permutations and Combinations 10/37 Examples I I Given a set with n elements, what is P (n, r )? I Decompose using product rule: I What is number of permutations of set S ? What is the number of 2-permutations of set {a, b, c, d , e}? I I How many ways to pick first element? I How many ways to pick second element? I How many ways to pick i ’th element? I How many ways to pick last element? Thus, P (n, r ) = n · (n − 1) . . . · (n − r + 1) = I How many ways to select first-prize winner, second-prize winner, third-prize winner from 10 people in a contest? I n! (n−r )! I Salesman must visit 4 cities from list of 10 cities: Must begin in Chicago, but can choose the remaining cities and order. I How many possible itinerary choices are there? I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 11/37 Instructor: Işıl Dillig, 2 CS311H: Discrete Mathematics Permutations and Combinations 12/37 Combinations I Number of r -combinations An r-combination of set S is the unordered selection of r elements from that set I I The number of r -combinations of a set with n elements is written C (n, r ) I C (n, r ) is often also written as I I Theorem: Unlike permutations, order does not matter in combinations I Example: What are 2-combinations of the set {a, b, c}? I For this set, 6 2-permutations, but only 3 2-combinations n r CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig, 13/37 Proof of Theorem What is the relationship between P (n, r ) and C (n, r )? I Let’s decompose P (n, r ) using product rule: I First choose r elements I Then, order these r elements I How many ways to choose r elements from n? I How many ways to order r elements? I Thus, P (n, r ) = C (n, r ) ∗ r ! I Therefore, , read ”n choose r” n r CS311H: Discrete Mathematics = n! r ! · (n − r )! Permutations and Combinations 14/37 I We need to create a team with 5 members out of 10 candidates. How many different teams are possible? I When creating a team, we don’t care about order in which players were picked. Thus, we want C (n, r ), not P (n, r ) I I How many hands of 5 cards can be dealt from a standard deck of 52 cards? I P (n, r ) n! C (n, r ) = = r! (n − r )! · r ! CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig, 15/37 A Corollary I Examples I Instructor: Işıl Dillig, n r is also called the binomial coefficient C (n, r ) = Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 16/37 Another Example Corollary: C (n, r ) = C (n, n − r ) I There are 9 faculty members in a math department, and 11 in CS department. I If we must select 3 math and 4 CS faculty for a committee, how many ways are there to form this committee? I I I I I I I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 17/37 Instructor: Işıl Dillig, 3 CS311H: Discrete Mathematics Permutations and Combinations 18/37 More Complicated Example I One More Example I How many bitstrings of length 8 contain at least 6 ones? I I I I I I I I I I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 19/37 Instructor: Işıl Dillig, Binomial Coefficients I Recall: C (n, r ) is also denoted as n r and is called the I Binomial is polynomial with two terms, e.g., (a + b), (a + b)2 I n called binomial coefficient b/c it occurs as r coefficients in the expansion of (a + b)n Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 20/37 An Example binomial coefficient CS311H: Discrete Mathematics Permutations and Combinations I Consider expansion of (a + b)3 I (a + b)3 = (a + b)(a + b)(a + b) I = (a 2 + 2ab + b 2 )(a + b) I = (a 3 + 2a 2 b + ab 2 ) + (a 2 b + 2ab 2 + b 3 ) I = 1a 3 + 3a 2 b + 3ab 2 + 1b 3 1 Instructor: Işıl Dillig, 21/37 The Binomial Theorem I How many bitstrings of length 8 contain at least 3 ones and 3 zeros? 3 3 CS311H: Discrete Mathematics 1 Permutations and Combinations 22/37 Example Let x , y be variables and n a non-negative integer. Then, (x + y)n = n X n x n−j y j j I What is the expansion of (x + y)4 ? j =0 I I To get such a term, must choose n − j of the x ’s and j y’s I Once you pick n − j x ’s, # of y’s determined (and vice versa) I Since there is Instructor: Işıl Dillig, I Proof: Each term in the expansion if of the form c · x n−j y j n n −j n coefficient c is j = CS311H: Discrete Mathematics n j 4 1 I ; I Therefore, = 4 3 =; 4 2 = (x + y)4 = 1x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4 ways to pick x n−j , Permutations and Combinations 23/37 Instructor: Işıl Dillig, 4 CS311H: Discrete Mathematics Permutations and Combinations 24/37 Another Example I Corollary of Binomial Theorem What is the coefficient of x 12 y 13 in the expansion of (2x − 3y)25 ? I I Binomial theorem allows showing a bunch of useful results. I Corollary: k =0 I I I I Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations Corollary: n P (−1)k n k I =0 Corollary: I CS311H: Discrete Mathematics Permutations and Combinations 27/37 n P 2k Instructor: Işıl Dillig, Pascal’s Triangle n k 26/37 Permutations and Combinations 28/37 = 3n CS311H: Discrete Mathematics Proof of Pascal’s Identity n’th row consists of n k Permutations and Combinations Instructor: Işıl Dillig, 5 n k −1 + Proof: n k −1 + n k Multiply first fraction by 29/37 = I for k = 0, 1, . . . n CS311H: Discrete Mathematics This identity is known as Pascal’s identity Pascal arranged binomial coefficients as a triangle n +1 k I I Instructor: Işıl Dillig, Permutations and Combinations I Instructor: Işıl Dillig, I = 2n CS311H: Discrete Mathematics k =0 I I n k One More Corollary k =0 I Instructor: Işıl Dillig, 25/37 Another Corollary I n P n k −1 + = k k n k n! n! + (k − 1)!(n − k + 1)! (k )!(n − k )! and second by n k = CS311H: Discrete Mathematics n−k +1 n−k +1 : k · n! + (n − k + 1)n! (k )!(n − k + 1)! Permutations and Combinations 30/37 Proof of Pascal’s Identity, cont. I I n k −1 + n k = Interesting Facts about Pascal’s Triangle k · n! + (n − k + 1)n! (k )!(n − k + 1)! Factor the numerator: (n + 1) · n! (n + 1)! n n + = = k −1 k (k )!(n − k + 1)! k ! · (n − k + 1)! But this is exactly n +1 k I What is the sum of numbers in n’th row in Pascal’s triangle (starting at n = 0)? I Observe: This is exactly the corollary we proved earlier! n X n = 2n k k =0 Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig, 31/37 Some Fun Facts about Pascal’s Triangle, cont. I I I Instructor: Işıl Dillig, I Earlier, when we defined permutations, we only allowed each object to be used once in the arrangement I But sometimes makes sense to use an object multiple times I Example: How many strings of length 4 can be formed using letters in English alphabet? I Since string can contain same letter multiple times, we want to allow repetition! I A permutation with repetition of a set of objects is an ordered arrangement of these objects, where each object may be used more than once Which of the theorems we proved today explains this fact? CS311H: Discrete Mathematics Permutations and Combinations 33/37 Instructor: Işıl Dillig, Number of Permutations with Repetition I 32/37 Numbers on left are mirror image of numbers on right Instructor: Işıl Dillig, I Permutations and Combinations Permutations with Repetitions Pascal’s triangle is perfectly symmetric I CS311H: Discrete Mathematics CS311H: Discrete Mathematics Permutations and Combinations 34/37 General Formula for Permutations with Repetition How many strings of length k can be formed using the 26 lower-case letters in the English alphabet? I P ∗ (n, r ) denotes number of r-permutations with repetition from set with n elements I Theorem: P ∗ (n, r ) = n r I Proof: n ways to select first element, n ways to select second, . . . , n ways to select r ’th element I By product rule, n r r-permutations with repetition are possible Decompose using product rule: I How many ways to choose first letter? I How many ways to choose second letter? I ... How many possible strings? CS311H: Discrete Mathematics Permutations and Combinations Instructor: Işıl Dillig, 35/37 6 CS311H: Discrete Mathematics Permutations and Combinations 36/37 Examples I How many ways to assign 3 jobs to 6 employees if every employee can be given more than one job? I How many different four-digit numbers can be formed from the digits 1, 2, 3, 4? I How many different 3-digit numbers can be formed from 1, 2, 3, 4, 5? Instructor: Işıl Dillig, CS311H: Discrete Mathematics Permutations and Combinations 37/37 7