I would have done like this
First give each kid - one ball of each color. There is only one way to do that.
After this we are left with 7R, 5G, 3B balls., correct?
Now we are free to distribute them in any way we like. Let's start with one color.
7R - can be distributed in 9!/7!2! ways = 36
5G - 7!/5!2! = 21 ways
3R - in 10 ways
Final answer - 7560 ways !