Combinatorics Through Guided Discovery, 2004a

6 1. What is Combinatorics?
and the sets B i into which we have divided it, we often call the sets B 1 , B 2 ,...,B m
the blocks of the partition. In this language, the sum principle says that
if we have a partition of a set S, then the size of S is the sum of the sizes
of the blocks of the partition.
The product principle says that
if we have a partition of a set S into m blocks, each of size n, then S has
size mn.
You’ll notice that in our formal statement of the sum and product pinciple we
talked about a partition of a finite set. We could modify our language a bit to cover
infinite sizes, but whenever we talk about sizes of sets in what follows, we will be
working with finite sets. So as to avoid possible complications in the future, let
us agree that when we refer to the size of a set, we are implicitly assuming the set
is finite. There is another version of the product principle that applies directly in
problems like Problem 5 and Problem 6, where we were not just taking a union of
m disjoint sets of size n, but rather m disjoint sets of size n, each of which was a
union of m ′ disjoint sets of size n ′ . This is an inconvenient way to have to think
about a counting problem, so we may rephrase the product principle in terms of a
sequence of decisions:
• Problem 11. If we make a sequence of m choices for which
• there are k 1 possible first choices, and
• for each way of making the first i −1 choices, there are k i ways to make
the ith choice,
then in how many ways may we make our sequence of choices? (You need
not prove your answer correct at this time.)
The counting principle you gave in Problem 11 is called the general product
principle. We will outline a proof of the general product pinciple from the original
product principle in Problem 80. Until then, let us simply accept it as another
counting principle. For now, notice how much easier it makes it to explain why we
multiplied the things we did in Problem 5 and Problem 6.
⇒
Problem 12. A tennis club has 2n members. We want to pair up the members
by twos for singles matches.
(a) In how many ways may we pair up all the members of the club? (Hint:
consider the cases of 2, 4, and 6 members.) (h)
(b) Suppose that in addition to specifying who plays whom, for each
pairing we say who serves first. Now in how many ways may we
specify our pairs?