When one of the parents has 'A' blood group and the other parent has…

The correct answer is Option (4) → Either 'A' or 'O'

The ABO blood group is controlled by the gene I, which has three alleles: I^A, I^B and i. I^A is dominant over i. So blood group A has the genotype I^AI^A or I^Ai, and blood group O has only ii.

Parent with O: ii → every gamete carries i.

Parent with A, genotype I^AI^A: gametes I^A only → every child is I^Ai, blood group A.

Parent with A, genotype I^Ai: gametes I^A and i → children I^Ai (A) and ii (O) in a 1 : 1 ratio.

Option (1) Only 'A' — Incorrect. This holds only when the A parent is I^AI^A. An I^Ai parent can also have a child with group O.

Option (2) Only 'O' — Incorrect. Every A parent passes on I^A to at least some children, so group A children are always possible.

Option (3) Both 'A' and 'O' — Incorrect. A child gets only one allele from each parent. One child has one blood group, never two at once.

Option (4) Either 'A' or 'O' — Correct. Each child is I^Ai (A) or ii (O). Groups B and AB are not possible, because neither parent carries I^B.

NCERT Extract:
Co-dominance – “… ABO blood groups are controlled by the gene I. The plasma membrane of the red blood cells has sugar polymers that protrude from its surface and the kind of sugar is controlled by the gene. The gene (I) has three alleles I^A, I^B and i. The alleles I^A and I^B produce a slightly different form of the sugar while allele i does not produce any sugar. Because humans are diploid organisms, each person possesses any two of the three I gene alleles. I^A and I^B are completely dominant over i, in other words when I^A and i are present only I^A expresses (because i does not produce any sugar), and when I^B and i are present I^B expresses.”