**Core Concept**
In the ABO blood group system, blood types are determined by the presence or absence of specific antigens on the surface of red blood cells. The ABO blood group antigens are encoded by a single gene, with three alleles: A, B, and O. The A and B alleles are codominant, while the O allele is recessive. This leads to the possible genotypes and phenotypes of A, B, AB, and O blood groups.

**Why the Correct Answer is Right**
In this question, the father has blood group 'A', which means he can be either AA or AO genotype. The mother has blood group 'AB', indicating she is AB genotype. The possible genotypes of their children are AO, BO, AB, and BB. Since the father can only contribute the A or O alleles, none of their children will inherit the OO genotype, which is required to express the O blood group. Therefore, the O blood group is not possible in any of their children.

**Why Each Wrong Option is Incorrect**
**Option A:** A blood group is possible in their children if the father is AA genotype and the mother is AB genotype, resulting in the AO genotype in their offspring.
**Option B:** B blood group is possible in their children if the mother is AB genotype and the father is AO genotype, resulting in the BO genotype in their offspring.
**Option D:** AB blood group is possible in their children if the mother is AB genotype and the father is AO genotype, resulting in the AB genotype in their offspring.

**Clinical Pearl / High-Yield Fact**
In the ABO blood group system, the presence of the A or B antigen on red blood cells is crucial for transfusion compatibility. Understanding the genotypes and phenotypes of the ABO blood group system is essential for safe blood transfusions and preventing hemolytic reactions.

**✓ Correct Answer: C. O**