Why do three nucleotides code for one amino acid? Why not 5 nucleotides?
The more bases there are per codon the more information you can code for. There are only 22 different amino acids, in consequence we need minimum 3 bases per codon.


1 base-codon --> 4^1 = 4 possible codes which are:


A / T / C / G 

2 base-codon --> 4^2 = 16 possible codes which are:


AA / AT / AC / AG / TT / TA / TC / TG / CC / CA / CG / CT / GG / GC / GT / GA

3 base-codon --> 4^3 = 64 possible codes which are:


AAA / AAT / AAC / AAG / ACA / ACT / ACG / ACC / AGA / AGT / etc...

4 base-codon --> 4^4 = 256 possible codes which are:


AAAA / AAAT / AAAC / AAAG / AATA / AATC / AATG / etc..

5 base codon --> 4^5 = 1024 possible codes


If we had only 2 bases per codon we couldn't code for all of the possible different amino-acids.

On the other hand there are several costs associated with carrying lots of DNA. It takes space, it takes energy and matter to get copied, it takes time to copy it (generation time might be limited by copying time), … I have just asked this question hoping to make things clear about the possible costs. If there is a cost associated with having a 4-base codon system compare to a 3-base codon system but there is no associated benefit. Such trait (that has a cost with no benefit) are not expected to evolve (counter-selected).

You'll notice that there are 64-22=42 extra possibilities. In consequence the genetic code is redundant. Meaning that several codon codes for the exact same amino-acid. And interestingly, it is often a the last of the three bases of the codon the does no yield to any change in the coding sequence as you can see here.



This post about the evolution of genetic code might interest you.