I for one find this question hard to follow.

Here is the way I read it.

There is a increasing sequence of integers: .

There are sets .

We pick a number from , then we pick adifferent numberfrom , etc until we pick yet adifferent numberfrom . Thus we have different selections.

The question seems to be, "How many different selections are possible?"

Note the the number of integers in each set is

If this is the correct setup, the answer is: