There is a lot of combinatorics involved in studying DNA and gene sequencing. As you know, DNA is built of building blocks called nucleotides. There are $4$ different chemicals$($bases$)$ that make up the strands$/$strings of DNA: thymine, cytosine, adenine, and guanine $($T$,$C$,$A$,$ and G for short$).$ In these strings, order matters, so the string CT is different from the string TC$.$

$($a$)$ A base pair is a $2-$element string of DNA, where each base is one of the four chemicals T$,$C$,$A$,$ and G$.$ For example, AG$,$ TT$,$ and GA are all base pairs. How many such base pairs exist?

$($b$)$ How many different strings consisting of $5$ bases are there?

$($c$)$ Suppose that a mouse$$s hair color is coded with a DNA string. How long does a string of a DNA need to be if there are $200$ distinct hair colors that are possible for a mouse?