In: Statistics and Probability
If each of n balls is placed at random in k urns, what is the probability that exactly two urns remain empty?
Provided k<n
Total no. of ways each of n balls can be placed at random in k urns = kn
Now consider that only the first two cells remain empty.
This can be achieved by first putting one ball in each of the remaining k-2 urns, and then placing the remaining n-(k-2)=n-k+2 balls in the k-2 urns in (k-2)n-k+2 ways.
This event can happen for any two urns, which can be selected in ways.
Total no. of ways each of n balls can be placed at random in k urns in such a way that exactly 2 urns remain empty