Suppose that for a hash function h, its input message m is always 16 bits long and the hash output digest is fixed 6 bits long.

(1) Q1: How many distinct outputs exist?

(2) Q2: How many distinct inputs exist?

(3) For a given hash function H, let C denote the maximum number of collisions for any output (that is, the maximum size of any set of inputs that produce the same output). For example (note that this example uses different parameter settings for illustration of C, please ignore the parameter settings in answering the following questions), suppose that h(m) is 2 bits long, so there are 4 possible hash outputs (00, 01, 10, 11). Assume that the output 00 has 2 collisions (i.e., two inputs are hashed to 00), 01 has 3 collisions, 10 has no collision, and 11 has 4 collisions. Then, it holds that C=max{2, 3, 0, 4}=4. Q3: What is the minimum possible value of C for the hash function h?

(4) Q4: What is the maximum possible value of C for the hash function h?

please answer all four parts, I'll upvote if you did so