Math 100
Induction and Counting
Martin H. Weissman
1.
Basic Counting
1.1.
The Handshake Problem.
Suppose that forty people enter a room. Each person shakes each other
person’s hand, exactly once. How many handshakes took place? What if there were a hundred people? Can
you find a formula, for the number of handshakes, given
n
people?
1.2.
ChoiceSequences.
Suppose that we have a finite sequence of “choices” to make, and we wish to find
out how many “total choices” there are. We illustrate this with the example of “how to dress”:
•
I have four pairs of jeans in my drawer. I have to choose one to wear.
•
I have seven shirts in my closet. I have to choose one to wear.
•
I have two pairs of shoes. I have to choose one (pair) to wear.
How many ways are there of getting dressed? We use the following two
basic principles
:
•
If
n
is a positive integer, then the number of ways to choose 1 object from a collection of
n
objects
is equal to
n
.
•
If we make a final sequence of (independent) choices, then the total number of outcomes is equal to
the
product
of the numbers of choices at each step in the sequence.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Weissman
 Factorials, Counting, Natural number, Martin H. Weissman

Click to edit the document details