Find the number of ways to distribute $2$ identical balls to $8$ distinct boxes.
This is how I reasoned it:
As the boxes are distinct, the only way by which we could get different configurations would be by considering the number of ways in which $2$ distinct pairs of boxes can be chosen. This can be done in $\binom {8}{2} $ ways.
Do you think that I have reasoned it correctly?
It depends on what you mean by "distribute":
a) if you mean "(randomly) throw the balls into the boxes" , then it means that you consider equi-probable that at each launch you can choose one of the $8$ boxes, so a total of $8^2=64$ ways to do that;
b) if instead you mean "(randomly) pour the balls into the boxes" , meaning that you consider equi-probable any "occupation histogram" such as $(2,0, \cdots,0), \cdots,(1,1,0,\cdots,0), \cdots$, then that is equivalent to the number of weak compositions of $2$ into exactly $8$ parts, which is $\binom{2+8-1}{8-1} = 36$.