Long time no see...

My greetings to all forum users.

I know that if we have identical sets that each one of them has objects and we want to pick $k$ objects, one from each set, how many combinations do we have?

The formula is (stars and bars approach):

For example, we have two sets of four (same) primes and we want to find out how many combinations we can get, the answer is:

However if we have to pick up from sets, where every set is a subset of a previous one, how many combinations there are? For example, if we have the following 3 sets and pick up a prime from each one set, how many combinations do we have?

Is there a compact formula? An explanation, if possible would be thankful.

Thank you very much.