Step 1: Blood group inheritance follows Mendelian genetics. The genotypes are determined by the alleles inherited from the parents.
- Blood group A is represented by genotype \( I^A i \), where \( I^A \) is the A allele and \( i \) is the recessive O allele.
- Blood group B is represented by genotype \( I^B i \), where \( I^B \) is the B allele and \( i \) is the recessive O allele.
- Blood group AB is represented by genotype \( I^A I^B \), as both A and B alleles are expressed.
Step 2: Analyzing the given parental blood groups:
- Father has blood group B, so his genotype is \( I^B i \).
- Mother has blood group AB, so her genotype is \( I^A I^B \).
- The child has blood group A, meaning the child must have inherited the \( I^A \) allele. Therefore, the child must have genotype \( I^A i \), receiving \( I^A \) from the mother and \( i \) from the father.
Step 3: Conclusion:
Thus, the genotypes of the father, mother, and child are:
- Father: \( I^B i \)
- Mother: \( I^A I^B \)
- Child: \( I^A i \)
Hence, the correct answer is option (A).