$4 $ balls selected at random from $ 15$

Question

$15$ balls $ 4$ selected

$1$ blue, $2 $ green, $3$ red, $4$ white, $5$ yellow

What is the probability that $2$ are red and at least $1$ is white?

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Now the way this question is worded makes it seem as though you should account for the possibility of the case being $2$ red balls + $1$ white + $1$ other AND the possibility of $2$ red balls + $2$ white

So the answer is:

$$\frac{C_{3,2} C_{4,1}C_{8,1}}{C_{15,4}} + \frac{C_{3,2} C_{4,2}}{C_{15,4}}\,?$$

Answer 1

$\frac{\binom{3}{2}\binom{4}{1}\binom{8}{1}}{\binom{15}{4}}+\frac{\binom{3}{2}\binom{4}{2}}{\binom{15}{4}}$

So you are right