Question

Ten distinct balls are distributed among ten distinct boxes
1.find the probability that only one of these ten boxes is empty.
2.how would your answer change if the balls were indistinct while the boxes are distinct

Answer 1

No. of ways in which ten distinct balls can be distributed among ten distinct boxes =

1) If only one of the ten boxes is empty, we need to distribute 10 balls among 9 boxes and one of these 9 boxes should contain 2 balls and the remaining 8 boxes should contain 1 ball each

9 boxes can be selected from 10 boxes in 10C9 = 10 ways

Now, any one of the 9 boxes(for 2 balls) can be selected in 9 ways

2 balls can be selected in 10C2 = 45 ways for the box containing 2 balls

The remaining 8 boxes can be filled with remaining 8 balls such that each of the boxes gets 1 ball each in 8! ways

Thus, total number of ways = 10*9*45*8! = 45 * 10!

So, the required probability = = 0.0163

2) If the balls are indistinct while the boxes are distinct, the 10 balls can be distributed among the ten boxes(without restriction) in

= 19C9

Now, 10 identical balls are to be distributed among 9 distinct boxes such that none of these 9 boxes are empty.

This can be done in = 9C8 = 9 ways

Thus, the required probability = =