solving this question involves using the techiques and estimations from the Birthday Problem probability theory.

a$.$ Hash values are $256$ long, but assume that the only allowed hashes begin with $77$ zero bits. The current bitcoin blockchain has around $829,000$ blocks, so round that up to a million blocks. Estimate the probability that two blocks have the same hash value $($assume all but the $77$ fixed bits are random and uniformly distributed$).$

b$.$ Turning the problem around, How many blocks would need to be hashed before there$$s roughly probability $1/2$ of a repeated hash?

c$.$ The MD$5$ has function which produces $128-$bit hash values. Repeat the two calculations above for $128-$bit hash value $($still $77$ zero bits$).$