I have $\ 30 $ marbles. $\ 25 $ are white, $\ 3 $ are blue and $\ 2 $ are red. same color marbles are identical.

If I pick randomly and without replacement $\ 4 $ marbles, what is the probability that I'll pick two each two of two colors?

Trying to make it easier, I assumed all marbles are different, so there are $\ 30 \cdot 29 \cdot 28 \cdot 27 $ ways to pick them and then number of options for :

Picking $2$ blue and $2$ red marbles are $\ {3 \choose 2}{25 \choose 2} \cdot 4! $ options.

Picking $2$ blue and $2$ white marbles are $\ {3 \choose 2}{2 \choose 2} \cdot 4! $ options.

Picking $2$ white and $2$ red marbles are $\ {25 \choose 2}{2 \choose 2 }\cdot 4! $.

The three events are mutually exclusive, so I should be able to just add them all together but that's the wrong answer. Any suggestions?