We can redefine the above problem using definition of function
we have two set , A={1,2,...k} B={1,2,....n}
Now a hash function maps one element of set A to exactly one element of set B.
Total Mappings=No.of functions from set A to Set B=n^k
1. probability that bucket number 1 is empty after the Kth insertion
Total functions from set A to set B-{1}/All possible functions=(n-1)^k/n^k
2. probability that no collision has occurred in any of the K insertions
Total one-to-one function from set A to set B/All possible functions=P(n,k)/n^k ,P means permutation
3.probability that the first collision occurs at the Kth insertion
(Total one-to-one function from set A-{k} to set B)*(No of ways kth element can map such that collision occur)/All possible functions=p(n,k-1)*(k-1)/n^k