Bunuel wrote:
In a certain mathematical activity, we have five cards with five different prime numbers on them. We will distribute these five cards among three envelope: all could go in any envelope, or they could be broken up in any way among the envelopes. Then in each envelop, we find the product of all the cards in that envelope: that is the “number” of the envelope. An envelope containing no cards has the number 1. We then put the three envelope numbers in order, from lowest to highest, and that is our set. How many different sets can be produced by this process?
This is not a realistic GMAT problem, and its wording is deeply problematic. In math, a set is a collection of numbers that
is not in order. That's the definition of a set. If something is in order, it's a sequence. So a math question can never tell you to "put the three numbers in order, from lowest to highest, and that is our set", because you aren't making a "set" the instant you put things in order. All of the information about putting things in order is irrelevant here, but it naturally leads to the confusion expressed in some of the questions above, about what happens when one prime is much larger than the others.
I've also never seen a counting problem on the GMAT with anywhere close to as many cases as you need to consider when solving this one. I can't even recall an official GMAT problem where I needed to consider more than three cases, and those are very rare. Here, just identifying the cases requires some work, and then we end up with five of them (5/0/0, 4/1/0, 3/1/1, 3/2/0, 2/2/1), the last of which is a bit tricky to deal with (because you have to notice you need to divide by 2, since it doesn't matter in which order you pick the two pairs of primes).
The official solution is correct, if I'm correctly guessing what the question is trying to ask, but I wouldn't suggest any GMAT test takers be concerned if they either find the wording confusing or find the question difficult or time-consuming, since you won't see a question like this on the real test.