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

SERIES 195
might have considered other serial orders for
the arrangement of these men; for example,
according to their height or their weight.
These other suggested orders strike us as
trivial in connection with Prime Ministers,
and would not naturally occur to the mind;
but abstractedly they are just as good orders
as any other. When one order among terms
is very much more important or more obvious
than other orders, it is often spoken of as the
order of those terms. Thus the order of the
integers would always be taken to mean their
order as arranged in order of magnitude. But
of course there is an indefinite number of
other ways of arranging them. When the
number of things considered is finite, the
number of ways of arranging them in order
is called the number of their permutations.
The number of permutations of a set of n
things, where n is some finite integer, is
n X(n-1) X(n-2) X(n-3) X... X4X3 X X1
that is to say, it is the product of the first n
integers; this product is so important in
mathematics that a special symbolism is
used for it, and it is always written "n!"
Thus, 2!=2 X =2, and 3! =3 X X1 =6, and
4!=4X3X2X1 =24, and 5!=5X4X3X2X1
= 120. As n increases, the value of n! increases very quickly; thus 100! is a hundred
times as large as 99!