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CMSC 203 / 0201 Fall 2002

This week's topics include generalized combinations and permutations, permutations with repetition, and inclusion-exclusion principle. Homework #8 is due.

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CMSC 203 / 0201 Fall 2002

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

  2. TOPICS • (Probability theory cont.) • Generalized combinations and permutations • NOTE changes to syllabus: • Shifting of material; some chapter sections dropped; graphs (7.1-7.5) instead of Boolean algebra • NOTE topics on midterm: • 3.1-3.5: Proofs, induction, and program correctness • 4.1-4.6: Counting • 5.1, 5.3, 5.5-5.6: Recurrence relations; inclusion-exclusion • NOT chapters 6, 7, 10 (these will be on the final along with ALL EARLIER TOPICS)

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

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

  5. Concepts / Vocabulary • Permutations and combinations with repetition • “sampling with replacement” • Number of r-permutations of n objects with repetition = nr • Number of r-combinations of n objects with repetition = C(n+r-1, r) [D’Alembert’s method / bars and stars] • Table 4.6.1 gives formulas • Permutations with indistinguishable objecs • Theorem 3: Number of n-permutations of n objects, where there are ni objects of type i (i=1, …, k) = n! / (n1! n2! … nk!)

  6. Examples • Exercise 4.6.19: 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 4.6.27: How many different strings can be made form the letters in ABRACADABRA, using all the letters?

  7. Examples II • Exercise 4.6.35: 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 4.6.42: 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?

  8. FRI 11/8INCLUSION-EXCLUSION (5.5-5.6)

  9. 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 • |A1A2…An| = 1in|Ai| - 1i<jn|AiAj| - … + (-1)n+1 |A1A2…An| • Proof by mathematical induction…

  10. 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?

  11. 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?

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