I'm trying to understand the definition of cardinality of permutations through a basic example. If, for example, there is a set:
$A = \{2, 3, 4, 2, 1\}$
What is the cardinality of its permutations?
And now if I add another two sets:
$B = \{7, 8, 9\}\\ C = \{1,2\}$
Can I define the cardinality of permutations between the three sets?
Your vocabulary is not quite right but I think I know what you mean.
What you call the "cardinality of permutations" of $2,3,4,2,1$ is the number of different ways to write those digits in some order. One such way is $23421$, another is $12234$.
If all five digits were different then there would be $5! = 120$ ways. But think about the two $2$'s. Imagine for the moment that one of them is red and the other black. Then you could tell the difference between the two ways to write $23421$ depending on where the red $2$ appeared. So if you answered $120$ you would be counting those two as different when they are really the same. You have double counted. The correct answer is $60 = 5!/2$.
This logic generalizes. If there were three $2$'s you would count each arrangement $3! = 6$ times. So for example putting $A$ and $C$ together your list is $1,1,2,2,2,3,4$ and there are $7!/2!3!$ ways to arrange it.
Last note: you should not write $A$ as you have and call it a set since sets can't have repeated elements. It's a list, or a multiset.