CSE20 Lecture 13

# CSE20 Lecture 13 - CSE 20 Lecture 13 Analysis: Counting...

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1 CSE 20 Lecture 13 Analysis: Counting with Pigeonhole Principle CK Cheng, 3/2/2010

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2 3.3 Pigeonhole Principle Pigeonhole Principle: If n pigeonholes are occupied by n + 1 or more pigeons, then at least one pigeonhole is occupied by more than one pigeon. Remark: The principle is obvious. No simpler fact or rule to support or prove it. Generalized Pigeonhole Principle: If n pigeonholes are occupied by kn + 1 pigeons, then at least one pigeonhole is occupied by k + 1 or more pigeons.
3 Example 1: Birthmonth In a group of 13 people, we have 2 or more who are born in the same month. # pigeons # holes At least people born on the same month 13 12 2 or more 20 12 2 or more 121 12 11 or more 65 12 6 or more 111 12 10 or more kn+1 n k+1

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Example 2: Handshaking Given a group of n people (n>1), each shakes hands with some (a nonzero number of) people in the group. We can find at least two who shake hands with the same number of people. Proof:
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## This note was uploaded on 06/12/2011 for the course CS 1 taught by Professor Staff during the Fall '08 term at Cornell.

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CSE20 Lecture 13 - CSE 20 Lecture 13 Analysis: Counting...

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