Is there a general formula to represent the cardinality of a Cartesian product of n sets?

Question

For two sets, basically, $|A \times B|=|A| \times |B|$..

Suppose I'm given a random number of sets.

Which formula should I use? Or is there a specific method?

Using a program It's easy to calculate it by counting every element, but I really don't recognize a pattern...

Answer 1

Think again about the Cartesian product of sets.
In $A \times B$ there are $|B|$ elements for every element in $A$, which are all different from the rest. This is some intuition for $|A \times B| = |A| \times |B|$.
Use exactly the same reasoning for: $$|A \times B \times C|=|(A \times B) \times C|=|A \times B| \times |C|=|A| \times |B| \times |C|.$$
etc..

I believe this is even stronger than a "formula", or in fact it $is$ one.