If out of $\displaystyle 3n$ letters there are $\displaystyle n\ As,n\ Bs\ \mbox{and}\ n\ Cs$, show that the number of ways of selecting $\displaystyle r$ letters out of these is the same as selecting $\displaystyle (3n - r)$ letters out of them. If $\displaystyle n < r < 2n + 1$, show that the number of ways of selecting $\displaystyle r$ letters is given by $\displaystyle \frac{1}{2}(n + 1)(n + 2) + (r - n)(2n - r)$