Chapter
2
Counting
11
HAPTER 2
I. BASIC PRINCIPLES OF COUNTING
This chapter is concerned with “counting”; counting the number of elements in a given
set or in some specified subset of a given set, counting the possible outcomes of some
experiment, or counting the number of ways to perform a sequence of operations. Our
objective is to provide systematic approaches to, and methods for answering the question:
How many?
Examples 1.1:
1.
Let
}
,
,
{
c
b
a
A
=
. How many subsets of
A
are there?
2.
If we toss a penny and a nickel, how many different possible outcomes are there?
3.
An experiment consists of flipping a coin and then rolling a die. How many different
possible outcomes are there?
Solutions:
1. The subsets of { , , }
a b c
are:
∅
,
}
{
a
,
}
{
b
,
}
{
c
,
}
,
{
b
a
,
}
,
{
c
a
,
}
,
{
c
b
,
}
,
,
{
c
b
a
.
There are 8 subsets.
2. The possible outcomes are
HH
,
HT
,
TH
,
TT
where, for example,
HH
signifies “head on the penny” and “ head on the nickel”, etc.
There are
4
possible outcomes.
3. The coin comes up in one of two ways;
H
or
T
, the die comes up in any one of six
ways; 1, 2, 3, 4, 5, 6. The possible outcomes are
{
H1, H2, … , H6, T1, T2, …, T
6}. There
are 2 6 12
⋅ =
possible outcomes.
C
COUNTING
COUNTING
COUNTING
COUNTING