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Introduction
Pigeonhole Principle
Discrete Mathematics
Andrei Bulatov
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View Full Document Discrete Mathematics – Pigeonhole Principle
272
Pigeonhole Principle
If
m
pigeons occupy
n
pigeonholes and
m > n,
then at least one
pigeonhole has two or more pigeons roosting in it.
Discrete Mathematics – Pigeonhole Principle
273
Examples
Among any group of 367 (or more) people, there must be at least
two with the same birthday, because there are only 366 possible
birthdays
In any group of 27 English words, there must be at least two that
begin with the same letter, because there are 26 letter in the Latin
alphabet
A function
f
from a set with
k + 1
elements to a set with
k
elements is not onetoone
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274
More Examples
Recall that a number
c
is called the
remainder
of
a
when divided
by
n
if
 0
≤
c < n
 a = k
·
n + c
for some
k
Note that two numbers,
a
and
b,
have the same remainder
when divided by
n
if
a – b
is divisible by
n
Among any group of
n + 1
integers, there must be at least two
with the same remainder when divided by
n
Discrete Mathematics – Pigeonhole Principle
275
More Examples
For every integer
n
there is a multiple of
n
that has only zeros
and ones in its decimal expansion
Solution:
Let
n
be a positive integer.
Consider the
n + 1
integers
1,
11,
111,
… ,
11…1
(where the last integer in this list is the integer with
n + 1 ones in its
decimal expansion).
At least two of them have the same remainder when divided by
n.
Let they have the form
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This note was uploaded on 10/14/2011 for the course MACM 101 taught by Professor Pearce during the Spring '08 term at Simon Fraser.
 Spring '08
 PEARCE

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