The genotype B-group father of an O-group child is
\[\begin{align}
& A. {{I}^{O}}{{I}^{O}} \\
& B. {{I}^{B}}{{I}^{B}} \\
& C. {{I}^{A}}{{I}^{B}} \\
& D. {{I}^{O}}{{I}^{B}} \\
\end{align}\]

Hint: The blood grouping in humans is done by ABO method. I stand for the antigen present and A, B and O stands for the alleles. These alleles determine the inherited blood group in individuals.

Complete answer: Out of A, B and O alleles, A and B are dominant in nature and O is recessive. That is when inherited together, A and B are expressed together resulting in AB blood groups with the genotype ${{I}^{A}}{{I}^{B}}$. On the other hand, when A or B are expressed with O, the resulting blood groups will be A or B since the allele ${{I}^{O}}$ is recessive to ${{I}^{A}}$ and ${{I}^{B}}$. Thus, the blood group O occurs only if both the parents possess an ${{I}^{O}}$ allele and the child has a genotype ${{I}^{O}}{{I}^{O}}$. Therefore, the father with blood group B should possess a genotype containing an allele of ${{I}^{O}}$. Therefore the genotype of the father is ${{I}^{O}}{{I}^{B}}$.
a) An individual having genotype ${{I}^{O}}{{I}^{O}}$ can have only O blood group since both the alleles are same even though they are recessive in nature, no other dominant alleles are present. Thus. the father cannot have a B blood group.
b) ${{I}^{B}}{{I}^{B}}$genotype can have B blood group, but since the allele, ${{I}^{B}}$ is dominant over O allele, the offspring cannot have an O group. AB blood groups are devoid of an O allele in any of its genes, therefore, the child cannot have an O blood group.
Thus, the correct answer is option D.

Note: A variant form of a gene is known as an allele. Human beings being diploid organisms possess two alleles in each gene loci. One allele inherited from each parent which are passed on through gametes. Alleles that express themselves are dominant, whereas alleles that don’t are recessive.