The ABO blood group system is determined by the presence of antigens on the surface of red blood cells and is controlled by the gene I, which has three alleles: $I^A$, $I^B$, and $i$. The alleles $I^A$ and $I^B$ are codominant, while $i$ is recessive.
Given the blood types:
- Father's blood group $B^+$: Possible genotypes are $I^B I^B$ or $I^B i$.
- Mother's blood group $A^+$: Possible genotypes are $I^A I^A$ or $I^A i$.
- Child's blood group $O^+$: Genotype must be $ii$, as $O$ blood group requires two recessive alleles.
Let's verify each option:
- Option A: $I^B i/I^A i/ii$
- Father: $I^B i$
- Mother: $I^A i$
- Child: $ii$
- This combination is possible, as both $I^B i$ and $I^A i$ can produce an $ii$ offspring.
- Option B: $I^B I^B /I^A I^A /ii$
- Father: $I^B I^B$
- Mother: $I^A I^A$
- Child: $ii$
- This option is not possible. Two homozygous dominant parents ($I^B I^B$ and $I^A I^A$) cannot have an $ii$ child.
- Option C: $I^A I^B /iI^A /I^B i$
- Incompatible genotypes and results. Skipping explanation as it doesn't relate meaningfully to provided parent-offspring relationship.
- Option D: $I^A i/I^B i/I^A i$
- Incompatible genotypes since no $ii$ for $O^+$ child.
- Option E: $iI^B /iI^A /I^A I^B$
- Incompatible genotypes as $I^A I^B$ child cannot be $O^+$.
The only viable genotype combination that allows for an $O^+$ child is in Option A: $I^B i/I^A i/ii$.
Hence, the correct answer is A only.