Step 1: Understanding the Question:
The question asks us to determine which blood groups will be completely absent in the offspring of a homozygous B mother and a homozygous A father.
Step 2: Key Concept and Approach:
Blood groups in humans are determined by the ABO gene, which exhibits multiple alleles: $I^A$, $I^B$, and $i$.
The alleles $I^A$ and $I^B$ are codominant, whereas $i$ is recessive.
We will find the genotypes of both parents and use a Punnett square to determine the possible genotypes and phenotypes of their children.
Step 3: Detailed Explanation:
• Parental Genotypes: Since the mother has blood group B and is homozygous, her genotype is $I^B I^B$.
Since the father has blood group A and is homozygous, his genotype is $I^A I^A$.
• Gametes Produced: The mother produces gametes containing only the $I^B$ allele, while the father produces gametes containing only the $I^A$ allele.
• Offspring Genotype: Fertilization of these gametes results in offspring with the genotype $I^A I^B$.
• Offspring Phenotype: Due to codominance between $I^A$ and $I^B$, all children will express the AB blood group.
• Absent Blood Groups: Consequently, the blood groups A, B, and O will be completely absent in their children.
Step 4: Final Answer:
The blood groups absent in their children are B, O, and A, which corresponds to Option (C).