Pigeonhole Principle
Suppose there are 6 pigeons and only 5 pigeonholes. Then at least
one pigeonhole must contain 2 pigeons if all pigeons fly into
pigeonholes.
Pigeonhole Principle: A function from one finite set to a smaller
finite set cannot be 1 – 1; there must be at least two elements in
the domain that have the same image in the co-domain.
ex. Suppose four cards are drawn from a 52 card deck (A – 2,
). Must at least two be of the same suit?
Suppose you want to find the number to pick to ensure a result.
ex. How many cards must you pick from a standard 52 card deck
to be sure of getting at least one red card?
ex. Suppose six pairs of similar looking boots are thrown together
in a pile. How many individual boots must you pick to be sure of
getting a matched pair?
ex. Let T = {1, 2, 3, 4, …, 9}. Suppose five integers are chosen
from T. Must there be two integers whose sum is 10?
Generalized Pigeonhole Principle:
For any function f from a finite set X to a finite set Y and for any
positive integer k, if n(X) > k n(Y), then y  Y such that y is
the image of at least k + 1 elements.
ex. 6 pigeons, 5 holes

ex. 32 pigeons, 5 holes
ex. A programmer writes 500 lines of computer code in 17 days.
Must there have been at least one day when the programmer wrote
30 or more lines of code?
Contrapositive of the Pigeonhole Principle
For any function f from a finite set X to a finite set Y and for any
1
positive integer k, if for each y  Y, f (y) has at most k
elements, then X has at most k(n(Y)) elements.

ex. There are 42 students who are to share 12 computers. Each
student uses exactly 1 computer and no computer is used by more
than 6 students. Show that at least 5 computers are used by 3 or
more students.
Theorem 7.4.1: For any function f from a finite set X to a finite
set Y, if n(X) > n(Y), then f is not 1 – 1.
Theorem 7.4.2: Let X and Y be finite sets with the same number
of elements. Suppose f is a function from X to Y. Then f is 1 – 1
iff f is onto.
Do:
1. Prove that your bank statement for any 10 year period will have,
at least twice, exactly the same number of cents in its monthly
balance.
2. Show that for any 11 positive integers, the difference between
some two of them is divisible by 10. (Hint: consider the units
digit of the numbers with the pigeonholes being the digits 0, 1, 2,
…, 9.)
3. You have a square dartboard measuring one foot on a side.
You throw 9 darts, all of which hit the board. Why is it that at
least 3 of the darts cannot be separated from one another by more
2
than 2  .707 feet?
