Answer: A. A is correct. The father is IᴬIᴮ, so each child receives either Iᴬ or Iᴮ from him — never both, and never neither. The mother is ii, so every child receives an i from her. The only two genotypes available are therefore Iᴬi, which is group A, and Iᴮi, which is group B, in a 1:1 ratio. Note what this rules out at BOTH ends: a child cannot be AB, because that needs Iᴬ and Iᴮ from the two different parents and the mother has neither; and a child cannot be O, because that needs an i from each parent and the father has none.
Why B is wrong: B is wrong because it lists every group indiscriminately. AB and O are each impossible here for opposite reasons — AB needs an Iᴬ or Iᴮ from the mother, who has neither, and O needs an i from the father, who has none.
Why C is wrong: C is wrong because AB is the one outcome the father cannot produce on his own: he passes ONE of his two alleles, not both. A child of an AB parent is never AB unless the other parent also supplies Iᴬ or Iᴮ.
Why D is wrong: D is wrong because group O requires the genotype ii, meaning an i allele from each parent. The father is IᴬIᴮ and carries no i at all.