How do you mean that not all subsets of an infinite set are finite, or in what cases would this be so?

Take all the even numbers. That's a subset of all whole numbers. Is it a finite set? Clearly not. Both are infinite; the population of even numbers is 50%. Sometimes infinity/infinity = 0.5.

The set of prime integers is a strict subset of the set of integers, and yet both are infinite (they have the same cardinality even--that is, they are the same size, in one sense of 'size').

Ex: {n ∈ R ∣ n is odd}

The universe isn't infinite either. But neither of those is the major error with his reasoning.

Douglas Adams wasn't writing a treatise on mathematics...

He was not, but the usage of the quote in this context implies some relevance to mathematical averaging and it's flaws. If the reasoning is not sound, neither is the usefulness in this context.