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 .

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