CMSC 203 / 0201 Fall 2002

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CMSC 203 / 0201Fall 2002

• Week 11 4/6/8 November 2002
• Prof. Marie desJardins

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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)

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MON 11/4 (PROBABILITY THEORY CONT. (4.5))

• see week 9 notes

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WED 11/6GENERALIZED PERMUTATIONS AND
COMBINATIONS (4.6)

• HOMEWORK 8 DUE

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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(nr-1, r) DAlemberts 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 (i1, , k)
  n! / (n1! n2! nk!)

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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?

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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?

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FRI 11/8INCLUSION-EXCLUSION (5.5-5.6)
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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?nAi - ?1?iltj?nAi?Aj -
  (-1)n1 A1?A2??An
• Proof by mathematical induction

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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?

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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?