Input =16 bit ; so total no of inputs = 2^16 (but here in qurestion no use of this information )
8 bit hash value so size of hash table = 2^8 =256
now we have to insert 1000 value in 256 cells.
if 3 values maps to same hash cell then = 256*3 =768 ( means if one hash table cell is mapped to 3 input only 768 input are covered)
if 4 values maps to same hash cell then = 256*4 =1024 ( means if one hash table cell is mapped to 4 input than we can cover 1024 input. But we have only 1000 input)
So some hash table cell will mapped to 3 input and some hash table cell will be mapped to 4 inputs .
so 4 is the largest k such that in any set of 1,000 inputs, there are at least 4 inputs that h maps to the same hash value.
so ans is option (B).