Question

We have ? different boxes and ? different objects. We want to distribute the objects into the boxes such that at no box is empty. In how many ways can this be done? Justify your answer.

Answer 1

We have k different boxes and r different objects.

We want to distribute the objects into boxes, such that no box is empty.

Now, to keep no box empty, we choose k objects from r, whch can be done (r C k) ways.

These k objects can be arranged in k boxes, one each, in k! ways.

Now, there are (r-k) objets remaining. Each of them can be placed in any of the k boxes.

So, these (r-k) objects can be arranged in k^(r-k) ways.

So, the all possible number of ways in which it can be done, is

So, the all possible number of ways, in which r objects can be arranged in k different boxes, such that no box is empty, is