Sets counting

Hello,

I am planning to develop a computer system for storing large amounts
of data in some smart way.
The thing I need now is to estimate number of items I will need to
store.
The problem is as follows:
I have a specified number of sets - say A, B and C. Each set has its
own number of elements, e.g. A1, A2, B1, B2, B3, C1, C2. What I need
to count is the number of sets that are produced from these sets in a
way that the new set can have 0 or 1 item from each of the above sets.
So in the example it can be one element sets, like A1, B3 etc, two
element sets - A1B3, B2C1 etc, and three element sets - A1B3C2, A2B2C1
etc.
How to compute the number of sets that can be produced from N sets,
where each set has Ki number of items?

Thanks for your help
--
Michal

.

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