Combinatorics Through Guided Discovery, 2004a

1.3. Some Applications of the Basic Principles 25
1.3.3 The pigeonhole principle
◦ Problem 60. American coins are all marked with the year in which they
were made. How many coins do you need to have in your hand to guarantee
that on two (at least) of them, the date has the same last digit? (When we
say “to guarantee that on two (at least) of them,...” wemean that you can
find two with the same last digit. You might be able to find three with that
last digit, or you might be able to find one pair with the last digit 1 and one
pair with the last digit 9, or any combination of equal last digits, as long as
there is at least one pair with the same last digit.)
There are many ways in which you might explain your answer to Problem 60.
For example, you can partition the coins according to the last digit of their date;
that is, you put all the coins with a given last digit in a block together, and put
no other coins in that block; repeating until all coins are in some block. Then you
have a partition of your set of coins. If no two coins have the same last digit, then
each block has exactly one coin. Since there are only ten digits, there are at most
ten blocks and so by the sum principle there are at most ten coins. In fact with ten
coins it is possible to have no two with the same last digit, but with 11 coins some
block must have at least two coins in order for the sum of the sizes of at most ten
blocks to be 11. This is one explanation of why we need 11 coins in Problem 60.
This kind of situation arises often in combinatorial situations, and so rather than
always using the sum principle to explain our reasoning , we enunciate another
principle which we can think of as yet another variant of the sum principle. The
pigeonhole principle states that
If we partition a set with more than n elements into n parts, then at least
one part has more than one element.
The pigeonhole principle gets its name from the idea of a grid of little boxes that
might be used, for example, to sort mail, or as mailboxes for a group of people in an
office. The boxes in such grids are sometimes called pigeonholes in analogy with
stacks of boxes used to house homing pigeons when homing pigeons were used to
carry messages. People will sometimes state the principle in a more colorful way
as “if we put more than n pigeons into n pigeonholes, then some pigeonhole has
more than one pigeon.”
Problem 61. Show that if we have a function from a set of size n to a set of
size less than n, then f is not one-to-one. (h)
Problem 62.
• Show that if S and T are finite sets of the same size, then a
function f from S to T is one-to-one if and only if it is onto. (h)