I want to know the general form of the answer to this example.
How many different arrangements are there of 4 objects, 2 of type 1 and 2 of type 2, where there are 20 objects of type I and 30 objects of type II? All objects are distinguishable.
I tired: $P_{2}^{20}P_{2}^{30} $, among other things, but it doesn't yield the answer. Is that because the cartesian product doesn't account for all the arrangements?
How could I count this positively, meaning, I don't want to subtract the other possible arrangements to get to this one? I understand how to do that much.
Sorry if it's a dumb question, help would be much appreciated! thanks!
Assuming that the order matters: first your choose 4 objects (2 from each type) and then permute them, so that the expression is $$\binom {20}2\binom {30}24!. $$ If the order does not matter omit $4! $.