Step 1: Write down the parental genotypes.
A homozygous B mother is $I^B I^B$ and a homozygous A father is $I^A I^A$.
Step 2: Combine the gametes.
The mother can only give an $I^B$ allele and the father can only give an $I^A$ allele, so every child must be $I^A I^B$.
Step 3: Read off the phenotype.
Since $I^A$ and $I^B$ are codominant, every child expresses both antigens together and has blood group AB, and only AB, meaning A, B and O blood groups can never appear in this family.
Final answer: Option 3, blood groups B, O and A are absent.