The ABO blood group system is defined by red blood cell surface antigens, governed by the gene I with three alleles: $I^A$, $I^B$, and $i$. $I^A$ and $I^B$ are codominant, while $i$ is recessive.
Given the phenotypes:
- 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 $ii$, requiring two recessive alleles.
Evaluating each option:
- Option A: $I^B i/I^A i/ii$
- Father: $I^B i$
- Mother: $I^A i$
- Child: $ii$
- This is possible because $I^B i$ 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 is impossible. Two homozygous dominant parents ($I^B I^B$ and $I^A I^A$) cannot produce an $ii$ child.
- Option C: $I^A I^B /iI^A /I^B i$
- Incompatible genotypes and outcomes. Skipped due to lack of relevance to the parent-offspring relationship presented.
- Option D: $I^A i/I^B i/I^A i$
- Incompatible genotypes as no $ii$ genotype is possible for an $O^+$ child.
- Option E: $iI^B /iI^A /I^A I^B$
- Incompatible genotypes because an $I^A I^B$ child cannot have an $O^+$ blood group.
The sole viable genotype combination enabling an $O^+$ child is presented in Option A: $I^B i/I^A i/ii$.
Therefore, the correct answer is A only.