### Title page for etd-0625109-115205

URN etd-0625109-115205 Wei-cheng Hung No Public. This thesis had been viewed 5231 times. Download 0 times. Applied Mathematics 2008 2 Master zh-TW.Big5 Chinese Inclusion-exclusion and pigeonhole principles 2009-06-05 116 pigeonhole principle Ramsey theorem derangement inclusion-exclusion principle consecutive permutation Eulerâ€™s phi function complete graph combinations with repetition onto function rook polynomial nonnegative integer solutions In this paper, we will review two fundamental counting methods: inclusionexclusion and pigeonhole principles. The inclusion-exclusion principle considersthe elements of the sets satisfied some conditions, and avoids repeat counting by disjoint sets. We also use the inclusion-exclusion principle to solve the problems of Euler phi function and the number of onto functions in number theory, and derangement and the number of nonnegative integer solutions of equations in combinatorics. We derive the closed-form formula to those problems. For the forbidden positions problems, we use the rook polynomials to simplify the counting process. We also show the form of the inclusion-exclusion principle in probability, and use it to solve some probability problems.The pigeonhole principle is an easy concept. We can establish some sets and use the pigeonhole principle to discuss the extreme value about the number ofelements. Choose the pigeons and pigeonholes, properly, and solve problems by the concept of the pigeonhole principle. We also introduce the Ramsey theorem which is an important application of the pigeonhole principle. This theorem provides a method to solve problems by complete graph. Finally, we give some contest problems about the inclusion-exclusion and pigeonhole principles to show how those principles are used. Mong-Na Lo Huang - chair Mei-Hui Guo - co-chair Fu-Chuen Chang - advisor indicate not accessible 2009-06-25

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