Question

number of distributions of four identical balls into three different boxes ​

Answer 1

Total ways to distribute 4 distinct balls into 3 boxes = 3*1 + 6*4 + 3*6 + 3*6 = 63. Edit: If the balls were to be considered identical, the total of unique ways to distribute them would be 3+6+3+3 = 15, which is to say that distinctness of the balls permits an additional 48 unique ways of distribution.