Combinatorics Through Guided Discovery, 2004a

2 1. What is Combinatorics?
learned in a sequence of problems, and some are quite challenging. When you find
a problem challenging, don’t give up on it, but don’t let it stop you from going on
with other problems. Frequently you will find an idea in a later problem that you
can take back to the one you skipped over or only partly finished in order to finish
it off. With that in mind, let’s get started. In the problems that follow, you will
see some problems marked on the left with various symbols. The preface gives a
full explanation of these symbols and discusses in greater detail why the book is
organized as it is! Table 1.1.1, which is repeated from the preface, summarizes the
meaning of the symbols.
• essential
◦ motivational material
+ summary
⇒ especially interesting
∗ difficult
· essential for this section or the next
Table 1.1.1: The meaning of the symbols to the left of problem numbers.
1.2 Basic Counting Principles
◦ Problem 1. Five schools are going to send their baseball teams to a tournament,
in which each team must play each other team exactly once. How
many games are required? (h)
• Problem 2. Now some number n of schools are going to send their baseball
teams to a tournament, and each team must play each other team exactly
once. Let us think of the teams as numbered 1 through n.
(a) How many games does team 1 have to play in?
(b) How many games, other than the one with team 1, does team two
have to play in?
(c) How many games, other than those with the first i − 1 teams, does
team i have to play in?
(d) In terms of your answers to the previous parts of this problem, what
is the total number of games that must be played?
Problem 3.
• One of the schools sending its team to the tournament has to
send its players from some distance, and so it is making sandwiches for