<s>[INST] Probability of picking two marbles each from two colors when selecting $4$ marbles out of $30$ marbles 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?
 [/INST] Correct answer is $\frac{401}{9135}$. It is calculated as follows:$\frac{\binom{25}{2}*\binom{3}{2}}{\binom{30}{4}}+\frac{\binom{25}{2}*\binom{2}{2}}{\binom{30}{4}}+\frac{\binom{3}{2}*\binom{2}{2}}{\binom{30}{4}}$
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