What is the difference between selecting $3$ balls from $5$ balls and distributing $5$ balls into $3$ boxes?

Question

What is the difference between the two cases?

1. selecting $3$ balls from $5$ balls
2. distributing $5$ balls into $3$ boxes

Answer 1

In the second example, some balls must be in the same box, and there's the question of which ones, and whether they are three in on box and one in each of the other two, or three in one box and two in one of the other two, or two in each of two boxes and one in the third, and there may also be the question of which balls end up in which boxes. There are many more ways this could turn out than in the first question.

Answer 2

There are more ways to distribute $5$ balls into $3$ boxes.

Even if the boxes are indistinguishable (and if they are distinguishable, there are even more ways yet), you could for example choose $3$ of the $5$ balls to put into one box (which equals the number of ways in which you can select $3$ out of $5$ balls), but after that you still have a choice whether you put the $2$ remaining balls into one of the other boxes, or to put $1$ ball into each of the remaining two boxes.

Of course, there are also ways to distribute $5$ balls into $3$ boxes where none of the boxes end up with exactly $3$ balls (e.g. one box ends up with all $5$ balls) and so there are even more choices yet.

Answer 3

1. How many ways you can choose 3 balls from 5 balls =$\binom{5}{3}$

2. You can get different answer depending on the capacity of boxes.

   lets say if we can put any number of balls into boxes, we have the following cases (5,0,0)(4,1,0)(3,2,0)(2,2,1)(3,1,1) which gives different answer depending on whether the boxes can be consider identical or different.