**Core Concept**
The question is testing the understanding of the ABO blood group system, which is determined 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 results in four possible phenotypes: A, B, AB, and O.

**Why the Correct Answer is Right**
In this scenario, the father has blood group B and the mother has blood group AB. The mother's blood group AB is the result of her inheriting the A and B alleles from her parents. Since the father has only the B allele, he can only pass on the B allele to his children. Therefore, the children are unlikely to have blood group A, as they would require the A allele from the father, which is not possible.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it is possible for the children to have blood group AB, as the mother can pass on both the A and B alleles to her offspring.

**Option B:** This option is incorrect because it is possible for the children to have blood group B, as the father can pass on the B allele to his offspring.

**Option C:** This option is incorrect because it is possible for the children to have blood group O, as the father can pass on the O allele, which he may have inherited from one of his parents.

**Clinical Pearl / High-Yield Fact**
In the ABO blood group system, the presence of the A or B antigen on red blood cells determines the individual's blood group. Understanding the inheritance pattern of the ABO blood group system is crucial in transfusion medicine to ensure compatible blood is given to patients.

**Correct Answer: D. A.**