> How many ways can you re-order <f, a, t> ? > <f a t> > <f t a> > <a f t> > <a t f> > <t f a> > <t a f> > > 3! = 3 Factorial = 3X2X1 = 6 > > How many different ways can you order <1,2,3,4....> ? > > oo X oo-1 X oo-2 X .... X 3 X 2 X 1 > It's not well defined!
Not if you write it like this, but the number of permutations of a set is well defined even for infinite sets -- just like the number of all subsets of a set is defined even for infinite sets.
For finite and infinite sets, two sets are considered to have the same number of elements if there exists a bijection between them. For example, the set of even naturals has the same size as the set of all naturals (N), because f(n)=2*n is a bijection between naturals and even naturals. For the same reason, the set of all sets of naturals (P(N) or, equivalently, 2^N) has the same size as the set of real numbers (R): There is a bijection between sets of naturals and reals.
There is a bijection between permutations of N and subsets of N, so the set of permutations has the same size as the set of subsets.