An introduction to mathematics, by A. N. Whitehead.

196 INTRODUCTION TO MATHEMATICS
It is easy to verify in the case of small
values of n that n! is the number of ways
of arranging n things in order. Thus consider two things a and b; these are capable
of the two orders ab and ba, and 2! =.
Again, take three things a, b, and c; these
are capable of the six orders, abc, acb, bac,
bca, cab, cba, and 3!= 6. Similarly for the
twenty-four orders in which four things a, b,
c, and d, can be arranged.
When we come to the infinite sets of things
-like the sets of all the integers, or all the
fractions, or all the real numbers for instance
-we come at once upon the complications of
the theory of order-types. This subject was
touched upon in Chapter VI in considering
the possible orders of the integers, and of the
fractions, and of the real numbers. The
whole question of order-types forms a comparatively new branch of mathematics of
great importance. We shall not consider it
any further. All the infinite series which we
consider now are of the same order-type as
the integers arranged in ascending order of
magnitude, namely, with a first term, and
such that each term has a couple of nextdoor neighbours, one on either side, with the
exception of the first term which has, of
course, only one next-door neighbour. Thus,
if m be any integer (not zero), there will be
always an mth term. A series with a finite