Combinatorics

# Thus the number of permuta5ons with this paxern is

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Unformatted text preview: ve the same birthday? –  Here n=10 and m=365 (ignoring leap years) –  Thus, the probability that two will have the same birthday is So, less than 12% probability CSCE 235 Combinatorics 23 Pigeonhole Principle: Example A (1) •  Show that –  in a room of n people with certain acquaintances, –  some pair must have the same number of acquaintances •  Note that this is equivalent to showing that any symmetric, irreﬂexive rela5on on n elements must have two elements with the same number of rela5ons •  Proof: by contradic5on using the pigeonhole principle •  Assume, to the contrary, that every person has a diﬀerent number of 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༄ CSCE 235 Combinatorics 24 Pigeonhole Principle: Example A (2) •  Assume, to the contrary, that every person has a diﬀerent number of...
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