September1999 CMSC 203 / 0201 Fall 2002 Week #11 – 4/6/8 November 2002 Prof. Marie desJardins

September1999 October 1999 TOPICS  (Probability theory cont.)  Generalized combinations and permutations  NOTE changes to syllabus:  Shifting of material; some chapter sections dropped; graphs ( ) instead of Boolean algebra  NOTE topics on midterm:  : Proofs, induction, and program correctness  : Counting  5.1, 5.3, : Recurrence relations; inclusion- exclusion  NOT chapters 6, 7, 10 (these will be on the final along with ALL EARLIER TOPICS)

September1999 MON 11/4 (PROBABILITY THEORY CONT. (4.5)) …see week 9 notes

September1999 WED 11/6 GENERALIZED PERMUTATIONS AND COMBINATIONS (4.6) ** HOMEWORK #8 DUE **

September1999 October 1999 Concepts / Vocabulary  Permutations and combinations with repetition  “sampling with replacement”  Number of r-permutations of n objects with repetition = n r  Number of r-combinations of n objects with repetition = C(n+r-1, r) [D’Alembert’s method / bars and stars]  Table gives formulas  Permutations with indistinguishable objecs  Theorem 3: Number of n-permutations of n objects, where there are n i objects of type i (i=1, …, k) = n! / (n 1 ! n 2 ! … n k !)

September1999 October 1999 Examples  Exercise : Suppose that a large family has 14 children, including two sets of identical triplets, three sets of identical twins, and two individual children. How many ways are there to seat these children in a row of chairs if the identical triplets or twins cannot be distinguished from one another?  Exercise : How many different strings can be made form the letters in ABRACADABRA, using all the letters?

September1999 October 1999 Examples II  Exercise : How many ways are there to travel in xyz space from the origin (0,0,0) to the point (4,3,5) by taking positive unit steps in any of the three directions?  Exercise : A shelf holds 12 books in a row. How many ways are there to choose five books so that no two adjacent books are chosen?

September1999 FRI 11/8 INCLUSION-EXCLUSION ( )

September1999 October 1999 Concepts / Vocabulary  Inclusion-exclusion revisited…  |A  B| = |A| + |B| - |A  B|  Inclusion-exclusion generalized…  |A  B  C| = |A| + |B| + |C| - |A  B| - |A  C| - |B  C| + |A  B  C|  Principle of Inclusion-Exclusion  |A 1  A 2  …  A n | =  1  i  n |A i | -  1  i<j  n |A i  A j | - … + (-1) n+1 |A 1  A 2  …  A n |  Proof by mathematical induction…

September1999 October 1999 Examples  Exercise 5.5.9: How many students are enrolled in a course either in calculus, discrete math, data structures, or programming languages if there are 507, 292, 312, and 344 students in these courses, respectively; 14 in both calculus and data structures; 213 in both calculus and programming languages; 211 in both discrete math and data structures; 43 in both discrete math and programming languages; and no student may take calculus and discrete math, or data structures and programming languages, concurrently?

September1999 October 1999 Examples II  Sieve of Eratosthenes  Derangements: Example 5.6.4: If n people check their hats at a restaurant, and the claim checks are misplaced, what is the probability that nobody receives the correct hat?