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Why is genetic code a triplet code? Logical Explanation

The logic behind triplet genetic code or why is the genetic code a triplet code instead of a singlet or doublet?
There are four nitrogenous bases in our DNA. Upon transcription, bases in mRNA are (Adenine A, Guanine G, Uracil U and Cytosine C) and transcribed mRNA codes for proteins. As we already know there are 20 amino acids (essential amino acids) that are coded by these four bases. Let's us consider the following possibilities. 
Genetic code singlet doublet triplet logic diagram

Case I
Now if we consider the genetic code to be singlet then it would be impossible as there are only four bases and that cannot code for 20 amino acids.  If codon is a singlet code then it can code for 4 amino acids only.
Genetic code, possible permutations
Case II:
If the genetic code is a doublet code; that is, an amino acid is coded by 2-nitrogeneous bases on mRNA in a specific sequence, then it can form 16 codons (42=4x4=16), still not sufficient enough to code 20 amino acids, so the genetic code can’t be of two letters. 
Case III: 
If the genetic code is a triplet code, that is, an amino acid is coded by 3-nitrogeneous bases, and then it can form 64 codons (43=4x4x4=64). But we have only 20 amino acids so codons are in excess. Then the possibility is some amino acids may be coded by more than one triplet code.
 Case IV: 
If the genetic code is a 4 letter code, that is, an amino acid is coded by 4-nitrogeneous bases then it can form 256 codons (44=4x4x4x4=256). But we have only 20 amino acid which is way too more. So it can't be 4.
Final verdict:
So the best possibility is for the codon to have 3 nitrogenous bases or each codon is a triplet code. George Gamow (1954) postulated each codon is a triplet code and is encoding the 20 standard amino acids of proteins used by living cells. Moreover, all the codon are specific, they will code for a single amino acid. Several triplets have the same letters but in different sequences and these code for different amino acids. Later by in vitro synthesis, scientists elucidated the triplet codons for all twenty amino acids.

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