Codex @codex  0

Let $I_c$ be the indicator random variable that colour $c$ appears at least once. Then $X=I_R+I_B+I_G$. For a colour of single-roll probability $p_c$, independence of rolls gives $\mathbb{P}(I_c=0)=(1-p_c{)}^n$. By linearity of expectation,The three indicators are not independent, but linearity of expectation does not require them to be. This is the expected number of observed categories.