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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 40, Wednesday, December 10

This lecture covers counting methods using Venn diagrams and the Inclusion-Exclusion principle. Topics include sets, unions, intersections, and permutations. Examples illustrate how to count elements in different sets using these techniques. Students will explore problem-solving strategies for combinatorial challenges.

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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 40, Wednesday, December 10

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  1. MATH 310, FALL 2003(Combinatorial Problem Solving)Lecture 40, Wednesday, December 10

  2. 8.1. Counting With Venn Diagram, 8.2 Iclusion-Exclusion Formula • Homework (MATH 310#12W): • Read 8.2 • Do 8.1: all odd numbered problems • Do 8.2: all odd numbered problems

  3. Example 1: Students Taking Neither Language • 100 students • 50 French • 40 Latin • 20 Both • How many students take no language? • Answer: 30. 50 20 40 100

  4. Inclusion - Exclusion • Inclusion – Ecclusion is a generalization of the addition principle: In general, if the two setes are not disjoint: • This can be generalized to n sets.

  5. Inclusion-Exclusion Fromula • Let U be a universal set and let Ai, for i = 1,2, ..., n, be arbitrary subsets of S. Le C(n,k) denote the collection of the k-subsets of the set {1,2,...,n}. let S(k) be the sum: • Then:

  6. Inclusion-Exclusion – Second Form • For any sets we have (A  B)c = AcBc: This implies: • Define S(0) = |U|. Therefore:

  7. Counting Derrangements • Let Dn denote the number of derrangements of n elements. (= permutations with no fixed points) • Theorem: • Dn = n!(1 – 1/1! + 1/2! – 1/3! + ... ±1/n!) • Proof: Inclusion - Exclusion.

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