Exercise
37
−
39
can be generalized as follows: If
0
≤
k
≤
n
and the set
A
has
n
elements, then the number of elements of the power set
P
(
A
)
containing exactly
k
elements is
(
n
k
)
.
Use this result to write an expression for the total number of elements in the power set
P
(
A
)
.
Use the binomial theorem as stated in Exercise 25 to evaluate the expression in part a and compare this result to exercise 27 and 37. (Hint: set
a
=
b
=
1
in the binomial theorem.)
If
n
is a nonnegative integer and the set
A
has
n
elements, then the power set
P
(
A
)
has
2
n
If
n
≥
2
and the set
A
has
n
elements, then the number of elements of the power set
P
(
A
)
containing exactly two elements is
(
n
2
)
=
n
(
n
−
1
)
2
.
If
n
≥
3
and the set
A
has
n
elements, then the number of elements of the power set
P
(
A
)
containing exactly three elements is
(
n
3
)
=
n
(
n
−
1
)
(
n
−
2
)
3
!
.
Let
a
and
b
be a real number, and let
n
be integers with
0
≤
r
≤
n
. The binomial theorem states that
(
a
+
b
)
n
=
(
n
0
)
a
n
+
(
n
1
)
a
n
−
1
b
+
(
n
2
)
a
n
−
2
b
2
+
...
+
(
n
r
)
a
n
−
r
b
r
+
.......
+
(
n
n
−
2
)
a
2
b
n
−
2
+
(
n
n
−
1
)
a
b
n
−
1
+
(
n
n
)
b
n
=
∑
r
=
0
n
(
n
r
)
a
n
−
r
b
r
Where the binomial coefficients
(
n
r
)
are defined by
(
n
r
)
=
n
!
(
n
−
r
)
!
r
!
,
With
r
!
=
r
(
r
−
1
)
.........
(
2
)
(
1
)
for
r
≥
1
and
0
!
=
1
. Prove that the binomial coefficients satisfy the equation
(
n
r
−
1
)
+
(
n
r
)
=
(
n
+
1
r
)
for
1
≤
r
≤
n