A set 
whose elements can be numbered through
from 1 to 
,
for some positive integer 
.
The number 
is called the cardinal number of the set, and
is often denoted 
or 
. In other words, 
is equipollent to the set
. We simply say that 
has 
elements. The empty set is also considered as a finite
set, and its cardinal number is 0.
A finite set can also be characterized as a set which is not infinite, i.e., as a set which is not equipollent to any of its proper subsets. In fact, if 
, and 
, a certain number 
of elements of 
do not belong to 
, so that 
.
For all 
,
the number of subsets having exactly 
elements (the so-called k-subsets,
or combinations of 
elements out of 
) is equal to the binomial
coefficient
 |  |  |
(1)
|
 |  |  |
(2)
|
Hence the number of subsets of 
(i.e., the cardinal number
of its power set) is
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
by virtue of the binomial theorem.
Assigning to each k-subset of 
its complement set defines
a one-to-one correspondence between the set of k-subsets
and the set of 
-subsets
of 
. This proves the identity
 |
(6)
|
The possible arrangements of the elements of 
are called permutations of
order 
. They all give rise to the same finite
ordinal 
,
since they are essentially the same; they can be transformed into each other simply
by renaming the elements. The number of permutations
of order 
is
 |
(7)
|
This is called 
factorial. In fact, when constructing an arrangement by
placing the elements in 
given positions, there are exactly 
possible choices for the first element, there are 
left for the second, and so on.
With respect to this notation, the number of combinations of 
elements can be written as
 |
(8)
|
The elements of each k-subset give rise to 
different permutations, so that the
total number of possible permutations of 
elements out of 
is
 |
(9)
|
