3 cubes Painted

The way I see it:
1. You've got three cases of "mono-color": RRR, GGG, BBB
2. Think of the case when you've got two cubes of the same color and the other one of a different color: there will be 3*2 = 6 such cases, since you've got three colors in total and two other options once you've already picked the other one
3. Last but not least: RGB or all the colors case

This makes 2 + 6 + 1 = 10 IMHO.
There might be an easier way with combinatorics/permutations, but I prefer to avoid this when I can, since I HATE them....

Try Different colors first.
RGB

The Try for Doubles + 1
RR (R+G+B)
BB (R+G+B)
GG (R+G+B)

total = 10