"It is so sweet to have chocolates on St. Valentine's Day!" Little Facer was so excited when he receives a box of self-made chocolates from his girlfriend, and he decided to eat these chocolates in a special way to commemorate this important day. Suppose that there are N
types of chocolates in the box, it can be easily calculated that there are totally
different combinations if we choose 3 chocolates of different types to make a dish. Facer will first make one dish for every of these combination, so there will be
dishes in all with 3×
chocolates totally. Then both Facer and his girlfriend choose some chocolates of different types as their original chocolates set. After that, Facer choose exactly M
dishes of three-type-mixed-chocolates which he made in the first step and add them into his original chocolates set. Finally, Facer continues eatting two chocolates of the same type until he cannot find any pair of chocolates of the same type (which means for each type of chocolates, there remains at most one chocolate). Facer wishes that his remaining chocolates set could be identical with his girlfriend's original chocolates set after the steps mentioned above, but he does not know the number of ways to choose dishes. Could please tell him the answer?