At last there is some problem which I can do.

Let be the number of red,blue,green balls selected.

Number of solutions to the equation ,

=coefficient of in

=coefficient of in

=coefficient of in (ignoring higher powers in the expansion)

=coefficient of in coefficient of

=

=

(i) which is clearly a multiple of 6.

(ii) which is clearly a multiple of 3.

But the real challenge would be to explain by using combinatorial argument.