Step 1: The O blood group is recessive and requires the genotype $ii$. This means both parents must contribute an '$i$' allele.
Step 2: A father with A blood group could be $I^A I^A$ or $I^A i$. A mother with AB blood group has the genotype $I^A I^B$.
Step 3: Since the mother only possesses $I^A$ and $I^B$ alleles, she cannot pass on the recessive '$i$' allele. Therefore, it is impossible for the child to have the $ii$ genotype (O group).