Question

In how many ways can we distribute 10 distinct balls into 5 distinct boxes with exactly 2 boxes empty?

Answer 1

We have to find the number of ways in which we can distribute 10 distinct balls into 5 distinct boxes, with exactly 2 boxes empty.

Now, both the balls and the boxes are distinct.

First, we chhose which two boxes, out of the 5, would remain empty.

This can be done in

ways.

Now, for each of these choices, we have to distribute the 10 distinct balls into the remaining 3 distinct boxes, such that none of them remains empty.

Now, to ensure none of these 3 remain empty, we first put one ball each in each box.

These 3 balls can be selected in

ways

and they can be arranged in the 3 boxes, in 3! number if ways.

Now, the remaining 7 balls can each go anywhere out of the 3 boxes, so each has 3 options, and they can be arranged in

number of ways.

So, the all possible ways is

So, the number of ways, in which we can distribute 10 distinct balls into 5 distinct boxes, such that exactly two remains empty, is 15,746,400.