**Core Concept**
The genetic code is a set of rules used by living cells to translate information encoded within genetic material (DNA or RNA sequences) into proteins. The genetic code is composed of codons, which are sequences of three nucleotides that correspond to a specific amino acid or a stop signal. In this question, we are dealing with synonymous codon pairs, which are codons that encode the same amino acid.

**Why the Correct Answer is Right**
AUG and AUA are not synonymous codon pairs because they encode different amino acids. AUG encodes methionine (Met), which is the start codon and the first amino acid in most proteins. AUA, on the other hand, encodes isoleucine (Ile). This is because AUG has a specific recognition site on the ribosome for the initiation of translation, whereas AUA is recognized by the tRNA molecule that carries isoleucine. This difference in recognition sites leads to the incorporation of different amino acids into the protein.

**Why Each Wrong Option is Incorrect**
**Option A:** CAU and CAC are synonymous codon pairs because they both encode histidine (His). CAU and CAC differ only in the third nucleotide, but both are recognized by the same tRNA molecule that carries histidine.
**Option B:** AUU and AUC are synonymous codon pairs because they both encode isoleucine (Ile). AUU and AUC differ only in the third nucleotide, but both are recognized by the same tRNA molecule that carries isoleucine.
**Option D:** AAU and AAC are synonymous codon pairs because they both encode asparagine (Asn). AAU and AAC differ only in the third nucleotide, but both are recognized by the same tRNA molecule that carries asparagine.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that the genetic code is degenerate, meaning that more than one codon can encode the same amino acid. This redundancy allows for the possibility of mutations in the genetic code without disrupting protein function.

**✓ Correct Answer: C. AUG & AUA**