Counting Chapter Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Counting Discrete Structures for Computing on 25 April 2011 Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Faculty of Computer Science and Engineering University of Technology - VNUHCM 6.1 Contents Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Introduction Contents Introduction Counting Techniques Counting Techniques Pigeonhole Principle Permutations & Combinations Pigeonhole Principle Permutations & Combinations 6.2 Introduction Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Example • In games: playing card, gambling, dices, • How many allowable passwords on a computer system? • How many ways to choose a starting line-up for a football match? Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.3 Introduction Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Example • In games: playing card, gambling, dices, • How many allowable passwords on a computer system? • How many ways to choose a starting line-up for a football match? Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations • Combinatorics (tổ hợp) is the study of arrangements of objects • Counting of objects with certain properties is an important part of combinatorics 6.3 Applications of Combinatorics Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan • Number theory • Probability Contents Introduction • Statistics Counting Techniques • Computer science Pigeonhole Principle • Game theory Permutations & Combinations • Information theory • 6.4 Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.5 Problems Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Contents • Number of passwords a hacker should try if he wants to use brute force attack (exhaustive key search) Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.6 Problems Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Contents • Number of passwords a hacker should try if he wants to use brute force attack (exhaustive key search) • Number of possible outcomes in experiments Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.6 Problems Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Contents • Number of passwords a hacker should try if he wants to use brute force attack (exhaustive key search) • Number of possible outcomes in experiments • Number of operations used by an algorithm Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.6 Product Rule Example Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan There are 32 routers in a computer center Each router has 24 ports How many different ports in the center? Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.7 Counting Combinations Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Definition (Combinations) An r-combination (tổ hợp chập r) of elements of a set is an unordered selection of r elements from the set Thus, an r-combination is simply a subset of the set with r elements Contents C(n, r) = n r n! = r!(n − r)! Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations Example How many ways are there to select eleven players from a 22-member football team to start up? C(22, 11) = 22! = 705432 11!11! 6.21 Exercises – Permutations with Repetition Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Suppose that a salesman has to visit eight different cities She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes How many possible orders can the salesman use when visiting these cities? Suppose that there are faculty members in CS department and 11 in CE department How many ways are there to select a defend committee if the committee is to consist of three faculty members from the CS and four from the CE department? Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.22 Permutations with Repetition Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Example How many strings of length r can be formed from the English alphabet? Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.23 Counting Permutations with Repetition Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Example How many strings of length r can be formed from the English alphabet? r By product rule, we see that there are 26 strings of length r Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.23 Counting Permutations with Repetition Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Example How many strings of length r can be formed from the English alphabet? r By product rule, we see that there are 26 strings of length r Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations Theorem The number of r-permutations of a set of n objects with repetition allowed is nr 6.23 Example Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Contents Question: How many ways we can choose students from the faculties of Computer Science, Electrical Engineering and Mechanical Engineering? Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.24 Counting Example Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan CCC CCE CCM CEE CMM CEM EEE EEM EMM MMM Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.25 Counting Example Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan CCC CCE CCM CEE CMM CEM EEE EEM EMM MMM || | | || | | || | | | | | | ??? || Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.26 Counting Example Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan CCC CCE CCM CEE CMM CEM EEE EEM EMM MMM || | | || | | || | | | | | | ??? || Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations How many ways to put and | ??? 6.26 Combinations with Repetition Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Theorem There are C(n + r − 1, r) r-combinations from a set with n elements when repetition of elements is allowed Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.27 Combinations with Repetition Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Theorem There are C(n + r − 1, r) r-combinations from a set with n elements when repetition of elements is allowed Contents Introduction Counting Techniques Example Pigeonhole Principle How many solutions does the equation Permutations & Combinations x1 + x2 + x3 = 11 have, where x1 , x2 , and x3 are nonnegative integers? 6.27 Examples Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Question: How many permutations are there of MISSISSIPPI? Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 6.28 Counting Examples Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Question: How many permutations are there of MISSISSIPPI? Contents Introduction Counting Techniques Pigeonhole Principle MISSISSIPPI ≡ MISSISSIPPI Permutations & Combinations 6.28 Permutations with Indistinguishable Objects Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Theorem The number of different permutations of n objects, where there are n1 indistinguishable objects of type 1, n2 indistinguishable objects of type 2, , and nk indistinguishable objects of type k, is Contents Introduction Counting Techniques n! n1 !n2 ! · · · nk ! Pigeonhole Principle Permutations & Combinations 6.29 Permutations with Indistinguishable Objects Counting Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan Theorem The number of different permutations of n objects, where there are n1 indistinguishable objects of type 1, n2 indistinguishable objects of type 2, , and nk indistinguishable objects of type k, is Contents Introduction Counting Techniques n! n1 !n2 ! · · · nk ! Pigeonhole Principle Permutations & Combinations Example How many permutations are there of MISSISSIPPI? 6.29