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The possible blood groups of children born to parents having A and AB groups are:
A). O, A
B). A, B, AB
C). O, A, B
D). O, A, B, AB

Answer
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Hint: Parents with A and AB blood group cannot have children with O blood group.

Complete answer:
Human blood group inheritance shows an example of codominance as well as multiple alleles.It is governed by three alleles: \[{I^A}\], \[{I^B}\] and \[{I^O}\]. \[{I^O}\] is recessive to both \[{I^A}\],and \[{I^B}\]. \[{I^A}\],and \[{I^B}\] may show codominance.
The parents have A and AB, blood groups. Blood group A indicates that the underlying genotype may be \[{I^A}{I^O}\] or \[{I^A}{I^A}\]. Blood group AB indicates co-dominance and the underlying genotype must be \[{I^A}{I^B}\]. Since we don't know what the genotype of the person with the blood group is, we have to take into account both cases.
In the first case (considering a person with A blood group has $\text{I}^\text{A}$$\text{I}^\text{O}$ genotype):
  \[{I^A}{I^O}\](A blood group) × \[{I^A}{I^B}\]. (AB blood group).

Gametes$\text{I}^\text{A}$$\text{I}^\text{O}$
$\text{I}^\text{A}$$\text{I}^\text{A}$$\text{I}^\text{A}$A blood group$\text{I}^\text{A}$$\text{I}^\text{O}$A blood group
$\text{I}^\text{B}$$\text{I}^\text{A}$$\text{I}^\text{B}$AB blood group$\text{I}^\text{B}$$\text{I}^\text{O}$B blood group


So, the possible $\text{I}^\text{B}$ blood groups of children born in this case are A, B, and AB blood groups.
In the second case (considering a person with A blood group has $\text{I}^\text{A}$$\text{I}^\text{A}$ genotype):
 \[{I^A}{I^A}\](A blood group) × \[{I^A}{I^B}\]. (AB blood group).

Gametes$\text{I}^\text{A}$$\text{I}^\text{A}$
$\text{I}^\text{A}$$\text{I}^\text{A}$$\text{I}^\text{A}$A blood group$\text{I}^\text{A}$$\text{I}^\text{A}$A blood group
$\text{I}^\text{B}$$\text{I}^\text{A}$$\text{I}^\text{B}$AB blood group$\text{I}^\text{A}$$\text{I}^\text{B}$AB blood group

So, the possible blood groups of children born in this case are A and AB blood groups.
In none of the above cases, children have O blood group.
Hence, the correct option is B). A, B, AB

Note:The ABO blood group system was discovered by American biologist Karl Landsteiner in 1901. Hence it is sometimes also called the Landsteiner blood group system.