$$A = \frac{ M! }{ r_1!\ r_2! }$$

where [itex] M = r_1 + r_2 [/itex],

where [itex] r_1 = (M - 2r_2) [/itex]

$$A = \frac{ (r_1 + r_2)! }{ r_1!\ r_2! } \\ \ \\ \

= \frac{ ((M-2r_2) + r_2)! }{ (M-2r_2)!\ (r_2)! } \\ \ \\ \

= \frac{ (M-r_2)! }{ (M-2r_2)!\ r_2! }

$$

Then, for a given M,

**A**maximum occurs at : [itex] r_2 = \frac{1}{2}r_1 [/itex].

I know this because I can generate a table algorithmically. But I'd like to know how to arrive here analytically. Any pointers would be appreciated.