Combinatorics

A combina5on is imply a subset of cardinality k csce

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Unformatted text preview: acquaintances: 0, 1, 2, …, n- 1 (no one can have n acquaintances because the rela5on is irreﬂexive). •  There are n possibili5es, we have n people, we are not done L༄ •  We need to use the fact that acquaintanceship is a symmetric irreﬂexive rela5on •  In par5cular, some person knows 0 people while another knows n- 1 people •  This is impossible. Contradic5on! J༄ So we do not have n (10) possibili5es, but less •  Thus by the pigeonhole principle (10 people and 9 possibili5es) at least two people have to the same number of acquaintances CSCE 235 Combinatorics 25 Pigeonhole Principle: Example B •  Example: Say, 30 buses are to transport 2000 Cornhusker fans to Colorado. Each bus has 80 seats. •  Show that –  One of the buses will have 14 empty seats –  One of the buses will carry at least 67 passengers •  One of the buses will have 14 empty seats –  Total number of seats is 80.30=2400 –  Total number of empty seats is 2400- 2000=400...
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