The gene that controls the ABO blood group system in human beings has three alleles-`I^(A), I^(B)` and i. A child has blood group O. His father has blood group A and mother has blood group (B) Genotypes of other off springs can be:
(I) `I^(B)I^(B)`
(II) `I^(A)i`
(III) `I^(B)i`
(IV) `I^(A)I^(B)`
(V) ii

Options

A. I, II, III, IV
B. II, III, IV, V
C. III, IV, V
D. IV, III, I

Explanation

To solve the question regarding the possible genotypes of the offspring from a father with blood group A and a mother with blood group B, while knowing that they have one child with blood group O, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the ABO Blood Group System**:
- The ABO blood group system is controlled by three alleles: `I^(A)`, `I^(B)`, and `i`.
- `I^(A)` and `I^(B)` are dominant alleles, while `i` is a recessive allele.

2. **Determine the Genotypes of the Parents**:
- The father has blood group A, which can be either `I^(A)I^(A)` (homozygous) or `I^(A)i` (heterozygous).
- The mother has blood group B, which can be either `I^(B)I^(B)` (homozygous) or `I^(B)i` (heterozygous).

3. **Analyze the Child's Blood Group**:
- The child has blood group O, which has the genotype `ii`.
- For the child to have blood group O, both parents must contribute an `i` allele. This means that the father must be `I^(A)i` and the mother must be `I^(B)i`.

4. **Determine Possible Genotypes of Other Offspring**:
- The possible combinations of alleles from the parents (`I^(A)i` from the father and `I^(B)i` from the mother) can be determined using a Punnett square:
- Possible combinations are:
- `I^(A)I^(B)` (Blood group AB)
- `I^(A)i` (Blood group A)
- `I^(B)i` (Blood group B)
- `ii` (Blood group O)

5. **List the Possible Genotypes**:
- The possible genotypes for the other offspring are:
- `I^(A)I^(B)` (AB)
- `I^(A)i` (A)
- `I^(B)i` (B)
- `ii` (O)

6. **Identify the Correct Options**:
- From the options provided:
- (I) `I^(B)I^(B)` - Not possible (both parents cannot provide `I^(B)`)
- (II) `I^(A)i` - Possible
- (III) `I^(B)i` - Possible
- (IV) `I^(A)I^(B)` - Possible
- (V) `ii` - Possible (but already accounted for the first child)

### Conclusion:
The possible genotypes of the other offspring can be:
- `I^(A)i`, `I^(B)i`, `I^(A)I^(B)`, and `ii`.