Section 2.6, Elementary Combinatorics
01/12/2010
1
2
Addition Rule for disjoint subsets of a finite set.
If {
A
1
,
A
2
, .
.. ,
A
n
} is a collection of (pairwise) disjoint
subsets of a finite set
S
, then
Elementary Combinatorics – Counting Tools
Definition.
Suppose that
A
is a finite set. The
cardinality
of
A
is the
number of elements in
A
, and is denoted 
A
.
Addition Rule for two subsets of a finite set.
If
A
and
B
are subsets of a finite set
S
, then
.
1
1
∑
=
=
=
n
k
k
n
k
k
A
A
U
.
B
A
B
A
B
A
∩
−
+
=
∪
3
Multiplication Rule.
Suppose a procedure consists of
m
steps, performed
sequentially, and that:
the 1
st
step can be performed in
n
1
ways
the 2
nd
can be done in
n
2
ways regardless of how the 1
st
was done
:
the
m
th
can be done in
n
m
ways regardless of the previous choices.
Then the number of different ways to perform the entire procedure is
n
1
n
2
···
n
m
.
Equivalently, the number of ways to fill
m
blanks, where:
the 1
st
blank can be filled in
n
1
ways
the 2
nd
can be filled in
n
2
ways regardless of how the 1
st
was filled
etc.
is
n
1
n
2
···
n
m.
Equivalently, the number of
m
tuples where the 1
st
coordinate can be
chosen in . . . .
4
.
1
1
∑
=
=
=
n
k
k
n
k
k
A
A
U
.
B
A
B
A
B
A
∩
−
+
=
∪
Multiplication Rule:
A procedure consists of
m
steps, performed sequentially, and for each
k
, step
k
can be performed in
n
k
ways regardless of the choices made on the previous
steps. Then the number of different ways to perform the entire procedure is
n
1
n
2
···
n
m
.
Addition Rule for two subsets of a finite set,
S
:
Addition Rule for disjoint subsets of a finite set,
S
:
Examples.
For each experiment, determine the number of possible outcomes.
Equivalently, determine the cardinality of the (natural or obvious) sample space for each.
1. Toss either a coin or a die, but not both.

S
 =
2. Toss both a coin and a die.

S
 =
3. Toss a die three times.

S
 =
5
.
1
1
∑
=
=
=
n
k
k
n
k
k
A
A
U
.
B
A
B
A
B
A
∩
−
+
=
∪
Multiplication Rule:
A procedure consists of
m
steps, performed sequentially, and for each
k
, step
k
can be performed in
n
k
ways regardless of the choices made on the previous
steps. Then the number of different ways to perform the entire procedure is
n
1
n
2
···
n
m
.
Addition Rule for two subsets of a finite set,
S
:
Addition Rule for disjoint subsets of a finite set,
S
:
Examples.
For each experiment, determine the number of possible outcomes.
Equivalently, determine the cardinality of the (natural or obvious) sample space for each.
4. Choose one card from a pile consisting of all the hearts and face cards
from a standard deck of cards.

S
 =
5. Arrange three people in a line (1
st
, 2
nd
, 3
rd
).

S
 =