In biology, a DNA is a double helix made of 2 sequences of nucleotides C, G, A and T. From computer science perspective, we can grossly oversimplify and view it as a pair ot two finite strings of the same length made out of letters C, G, A,T (here, we do not even care whether pairs of nucleotides match). For example, (TA, TT) would be a valid DNA of length 2 from our oversimplified perspective, even though it would make no sense to a biologist 

a. How many such DNA8 of length k ≥ 1 are there? Start by calculating the number of DNAs of length 1. Assume that order matters both for elements of a pair, and for letters in strings (so (CA, GT), (GT, CA), and (AC, TG) are three different DNAs) 

b. Show that the number of possible DNAS is countable. That is, show how to associate with every DNA a different natural number 

c. Suppose a computer scientist-turned-biologist defines a "species" as a (potentially infinite) set of possible DNAs. Show that with this definition the number of species is uncountable. Hint: you could do it by diagonalization, or you can, much simpler, directly infer it from one of the results that we stated when talking about diagonalization.