The ways of choosing spoken of have to do with what choices are made, in what order. They do not speak of whether the choice is made by entering an electronic record, shouting to a crowd, or with a quill pen. It is irrelevant how many bathroom breaks you take in your arduous task. It is all a question of what choices you made in what order.

There is one way to make no choices from 42 things. It matters not whether you describe this as failing to turn your paper in, or turning in a blank paper.

There are 42 ways to make one choice from 42 things. It matters not how many times you think of making no choices, they don't get recorded.

There are 1764 ways to make 2 choices with repetition allowed from 42 things. If you think of the operation as making a choice, making no choices 500 times, and then making your second choice, it matters not. The number of pauses is not relevant, and each time you don't choose you have only 1 way to do that. With this strange model you have complicated the overall calculation by multiplying by 1 500 times, which changes nothing.

I could go on, but I think the point is clear. From number theory to combinatorics to analytics it is widely accepted that an empty sum is 0 and an empty product is 1. This does not mess up the usual model of anything. In fact it is the usual model used by mathematicians...

Given a set containing N elements, how many different subsets are there that contain P elements? For P==0, the answer is always 1, the empty set.

Does that make it easier for you to understand? Update: Um, that sounds harsher than I intended it. For what it's worth, I found tilly's explanation hard to understand. I knew what he was getting at but the way he described there being only one possible choice of actions (no action) wasn't convincing to me, even leading me to thinking "there is no way to pick 0 things from 0 things because there is nothing to pick from". But changing terminology makes it very easy to understand, I think. The only choice I can make is to give you the empty set, the set that contains no items. So I'm not giving you zero sets, I am giving you one set.