Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
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2 1. What is <strong>Combinatorics</strong>?<br />
learned in a sequence of problems, and some are quite challenging. When you find<br />
a problem challenging, don’t give up on it, but don’t let it stop you from going on<br />
with other problems. Frequently you will find an idea in a later problem that you<br />
can take back to the one you skipped over or only partly finished in order to finish<br />
it off. With that in mind, let’s get started. In the problems that follow, you will<br />
see some problems marked on the left with various symbols. The preface gives a<br />
full explanation of these symbols and discusses in greater detail why the book is<br />
organized as it is! Table 1.1.1, which is repeated from the preface, summarizes the<br />
meaning of the symbols.<br />
• essential<br />
◦ motivational material<br />
+ summary<br />
⇒ especially interesting<br />
∗ difficult<br />
· essential for this section or the next<br />
Table 1.1.1: The meaning of the symbols to the left of problem numbers.<br />
1.2 Basic Counting Principles<br />
◦ Problem 1. Five schools are going to send their baseball teams to a tournament,<br />
in which each team must play each other team exactly once. How<br />
many games are required? (h)<br />
• Problem 2. Now some number n of schools are going to send their baseball<br />
teams to a tournament, and each team must play each other team exactly<br />
once. Let us think of the teams as numbered 1 through n.<br />
(a) How many games does team 1 have to play in?<br />
(b) How many games, other than the one with team 1, does team two<br />
have to play in?<br />
(c) How many games, other than those with the first i − 1 teams, does<br />
team i have to play in?<br />
(d) In terms of your answers to the previous parts of this problem, what<br />
is the total number of games that must be played?<br />
Problem 3.<br />
• One of the schools sending its team to the tournament has to<br />
send its players from some distance, and so it is making sandwiches for