TA’s Challenge Problem #6

(i) A box contains $\displaystyle 6n$ balls of three different colours: red, green, blue. There are an equal number of balls of each colour, and balls of the same colour are identical. If $\displaystyle C(n)$ denotes the total number of ways of selecting $\displaystyle 3n$ balls from the box, show that $\displaystyle C(n)-1$ is divisible by 6.

(ii) As previously, but now let $\displaystyle D(n)$ be the total number of ways of selecting $\displaystyle 3n$ balls such that there is at least one ball of each colour. Prove that $\displaystyle D(n)+2$ is divisible by 3.