C
OALITIONS
AND
C
AUCUSES
P
AGE
1
5/7/2009
Mathematical Background
A set of
n
objects contains exactly 2
n
subsets.
For
example, the
8
2
3
=
subsets of
{
}
c
b,
a,
are
{ }
{
}
{
}
{
}
{
}
{
}
{
}
{
}
.
c
b,
a,
,
c
b,
,
c
a,
,
b
a,
,
c
,
b
,
a
,
We have arranged the eight subsets in order of the number of elements they
contain: first the one subset of size zero, then the three subsets of size one, then
the three subsets of size two, and finally the one subset of size three.
Notice that
half of the subsets listed above contain at most two letters, and half contain at
least three letters.
In general, if
n
is an odd number, then exactly half of the 2
n
subsets of a set of
n
objects contain less than
2
n
objects, and half contain more than
2
n
objects.
So, for example,
10
2
of the
11
2
subsets of
{
}
k
j,
i,
h,
g,
f,
e,
d,
c,
b,
a,
contain at most
five letters, and the other
10
2
of them contain at least six letters.
In general, a set of size
n
contains exactly
(
29
(
29
(
29
(
29
(
29
(
29
!
!
!
1
2
1
1
2
1
factors
factors
k
k
n
n
k
k
k
k
n
n
n
n
k
n
k
k

=


÷
+



=
subsets of size
k
, where the symbol
n
! stands for the product of all the positive
integers from 1 to
n
.
For example, the number of subsets of
{
}
c
b,
a,
of size two
was seen to be
(
29
(
29
(
29
.
3
2
1
6
!
2
!
1
!
3
1
2
2
3
2
3
=
⋅
=
⋅
=
⋅
÷
⋅
=
Similarly,
(
29
(
29
.
120
!
7
!
3
!
10
1
2
3
4