Question

How to distribute (n+1) different balls into n boxes so that “no box will be empty”?

Answer 1

We have to distribute (n+1) different balls into n boxes, so that no box will be empty.

Now, no box would be empty, means we have to put at least one ball in each box.

Let us put one ball each in n boxes; first let us choose which n balls to put in n boxes, 1 each, out of these n+1 balls.

That can be done in ((n+1) C n) ways.

Now, this n balls can be put in n boxes, in n! number of ways.

The last ball can be put into any of the n boxes.

So, the number of favourable cases is

So, the number of ways in which (n+1) different balls can be distributed into n boxes, such that no box stays empty, is