The ABO blood group is determined by alleles I^A, I^B, and i, which are responsible for producing A, B, and O blood types, respectively. The I^A and I^B alleles are codominant, meaning both can be expressed if both are present, whereas the i allele is recessive.

Given the blood types in the problem statement:

• Father's blood group: $\mathrm{B}^{+}$ - Possible genotypes: I^BI^B or I^Bi
• Mother's blood group: $\mathrm{A}^{+}$ - Possible genotypes: I^AI^A or I^Ai
• Child's blood group: $\mathrm{O}^{+}$ - Possible genotype: ii

To find the correct pairing:

The child has an O blood type, meaning their genotype must be ii, indicating they inherited an i allele from each parent. Therefore, both parents must have at least one i allele.

Given this information:

• Father's Genotype: It must include i since the child inherited i. Thus, the father's genotype could be I^Bi.
• Mother's Genotype: It must also include i for the same reason, so the mother's genotype could be I^Ai.
• Child's Genotype: It is confirmed as ii.

Matching this analysis with the provided options:

• Option A: I^Bi/I^Ai/ii - This matches the reasoning provided.
• Option B: I^BI^B/I^AI^A/ii - This is incorrect because it suggests that neither parent has a recessive i allele to pass to the child, which is necessary for the child’s blood type O.
• Option C: iI^B/iI^A/I^AI^B - This is incorrect and does not match the required genotypes for the child to inherit ii.
• Option D: I^Ai/I^Bi/I^Ai - This includes a typo in the children's genotype and misalignment with blood types.
• Option E: iI^B/iI^A/I^AI^B - This is incorrect, mismatches the inheritance mechanism.

So the correct pairing based on the genotypes and blood groups mentioned, along with genetic laws, would be Option "A", i.e., I^Bi/I^Ai/ii.

So the answer is: Option A