Combinatorics

It is necessary to dis5nguish when the order maxers

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Unformatted text preview: pigeonhole, more than one pigeon must be in it •  Theorem: If k+1 or more objects are placed in k boxes, then there is at least one box containing two or more objects •  This principal is a fundamental tool of elementary discrete mathema5cs. •  It is also known as the Dirichlet Drawer Principle or Dirichlet Box Pinciple CSCE 235 Combinatorics 19 Pigeonhole Principle (2) •  It is seemingly simple but very powerful •  The difficulty comes in where and how to apply it •  Some simple applica5ons in Computer Science –  Calcula5ng the probability of hash func5ons having a collision –  Proving that there can be no lossless compression algorithm compressing all files to within a certain ra5on •  Lemma: For two finite sets A,B there exists a bijec5on f:A→B if and only if |A|=|B| CSCE...
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