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Wednesday's notes (1/30) MA204+MA284 : Discrete Mathematics Week 6: Advance applications of the PIE http://www.maths.nuigalway.ie/~niall/MA284 12 and 14 of October, 2016 1 “Stars and bars” 2 3 4 5 6 7 Multisets Problems with non-negative integer solutions Inequalities NNI equations with lower bounds on solutions Advanced Counting Using PIE Derangements Le probléme de rencontres General formula Miscellaneous Repetitions Permutations with indistinguishable objects Exercises See also Sections 1.5 and 1.6 of Levin’s Discrete Mathematics: an open introduction. Assignment 1 (2/30) Assignment 1 is now closed Your grades are available from the Blackboard Grade Centre. The average score was 17.3/20 (85%). ........................................................................... Which problem from Discrete Mathematics Assignment 1 would you like to study in more depth? I plan to provide more detailed, interactive notes for you on some of the topics covered in Discrete Mathematics Assignment 1. I will also use this material as part of my project work for a module on Learning Technologies (CEL263), that I am studying. I would be grateful if you would use the following is survey to indicate which material you would like developed further: https://goo.gl/forms/mz6HqIYwA1HhXfyS2 Assignment 2! (3/30) ASSIGNMENT 2 is now open! To access the assignment, go to https://webwork.nuigalway.ie/webwork2/1516-MA284 Your USERNAME is: Your PASSWORD is: There are 20 questions. You may attempt each one up to 10 times. This assignment contributes 10% to your final grade for Discrete Mathematics. Deadline: 5pm, Thursday, 3 November. For more information, see Blackboard, or http://www.maths.nuigalway.ie/~niall/MA284 “Stars and bars” (4/30) Last week we had the following question Every day you give some apples to your lecturers. Today you have 7 apples. How many ways can you give them to the 4 lecturers you have today? We learned that This is the same as finding the number of ways we can arrange 7 apples (stars), divided into 4 groups, separated by 3 bars. Any way can be written with 10 symbols (7 stars and 3 bars): we just � � ways. have to choose where to put the 3 bars. This can be done in 10 3 “Stars and bars” Multisets (5/30) Definition (Multiset) A multiset is a set of objects, where each object can appear more than once. As with an ordinary set, order does not matter. Examples: “Stars and bars” Multisets (5/30) Definition (Multiset) A multiset is a set of objects, where each object can appear more than once. As with an ordinary set, order does not matter. Examples: “Stars and bars” Multisets (5/30) Definition (Multiset) A multiset is a set of objects, where each object can appear more than once. As with an ordinary set, order does not matter. Examples: “Stars and bars” Multisets (6/30) How many sets of multisets of size 4 can you from the numbers {1, 2, 3, 4, 5}? “Stars and bars” Multisets (7/30) Example MA204 Semester 1 Examination 2014/15: Q2(a) 1. In how many ways can one distribute ten e1 coins to four students? 2. In how many ways can one distribute ten e1 coins to four students so that each student receives at least e1? “Stars and bars” Multisets (7/30) Example MA204 Semester 1 Examination 2014/15: Q2(a) 1. In how many ways can one distribute ten e1 coins to four students? 2. In how many ways can one distribute ten e1 coins to four students so that each student receives at least e1? Problems with non-negative integer solutions (8/30) A non-negative integer problem How many non-negative integer solutions are there to the problem x1 + x2 + · · · + xk = n? This is the same as... How many ways are there to distribute n identical objects among k individuals. The answer is � n+k −1 k −1 � = (n + k − 1)! n!(k − 1)! Problems with non-negative integer solutions Inequalities (9/30) Example (Part 1) 1. How many non-negative integer solutions are there to x1 + x2 + x3 = 3? Problems with non-negative integer solutions Inequalities (10/30) Example (Part 2) 2. How many non-negative integer solutions are there to x1 + x2 ≤ 3? Problems with non-negative integer solutions Inequalities (11/30) Looking at this example, it seems that The number of non-negative integer solutions to x 1 + x2 + x3 + · · · + x k ≤ n is the same as the number of non-negative integer solutions to x1 + x2 + x3 + · · · + xk + xk+1 = n. Why is that? Problems with non-negative integer solutions Inequalities (12/30) The number of NNI1solutions · + xk ≤ n, is exactly the 1 + x2 + 2 to x3 4 x3 + · · 5 same as the number of NNI solutions to x1 + x2 + x3 + · · · + xk < n + 1. Example (Q2(a), MA284, Semester 1 Exam, 2014/15) Find the number of non-negative integer solutions of the inequality x1 + x2 + x3 + x4 + x5 < 11 Answer: this is same as x1 + x2+ x 3+ x4 + x5 < 10 which is the same as x1 + x2+ x 3 + x4 + x5 + x =10 6 Finished here Wednesday