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

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 40, Wednesday, December 10. 8.1. Counting With Venn Diagram, 8.2 Iclusion-Exclusion Formula. Homework (MATH 310# 1 2W): Read 8.2 Do 8.1: all odd numbere d problems Do 8.2: all odd numbered problems.

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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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