How many recommendation letters should you ask a professor to write?

Normally, you should choose the two people who know you best and can write you the strongest letters, and ask each of them to send one letter to each program to which you are applying. Each of them will write one letter and send copies of it, with minor modifications if needed, to every program, so it isn't much more work than sending one letter.

This is to your advantage (you get the strongest letters) and also minimizes the total amount of work.

If a program needs more than two letters, or you think another professor would be a more suitable writer for one particular program (for instance, if they have connections to that institution), then you can ask for letters from them as well. But there's no need to get letters from more different people purely for the variety.

From my point of view as a recommendation writer, it's not much harder to send out 5 recommendation letters for a student then it is to send out one letter, since the letters vary only slightly and in most cases the letters can be submitted online. In my experience, when students ask me for recommendations I typically end up sending out four or five letters per student.

Thus I don't think that it is at all unreasonable to ask your recommendation writers to send out multiple letters (within reason.)

Your goal should be to get the absolute best letters possible.

Most likely, of the ten professors the candidate knows, there are two that will be 'better' letter-writers than the others - in the sense that they know her better, have taught her in multiple classes (more than the others) that she has done well in, the candidate may have done some research/reading course with them, etc.

In general, these two should be asked to write a letter each that they would send to the different programs. If a student were to come to me and ask me to send letters to five different programs, I would not write five entirely different letters. The letters might have some minor differences if the programs have some significant differences (applied vs theoretical program, or something like that), but they would be mostly identical.

The only situation where you might ask a third person is if they are a better choice for a particular program, e.g. they might be an alum of that program, or might know someone there personally, might have had the candidate in courses that are particularly relevant for that particular program, etc. In that case you would ask them to write a letter for that particular program in lieu of the default choices.