The correct answer is Option D: B and E. Here's why:

Let's break down the blood group system and genotypes:

• Blood Groups: The ABO blood group system is determined by three alleles: $I^A$, $I^B$, and $i$.
• Genotypes: The combination of these alleles determines the blood group. Here are the possible genotypes and their corresponding blood groups:
• • $I^A I^A$ or $I^A i$ = Blood group A
  • $I^B I^B$ or $I^B i$ = Blood group B
  • $I^A I^B$ = Blood group AB
  • $i i$ = Blood group O
• Rh Factor: The Rh factor is another blood group system. It's either positive (+) or negative (-). A person inherits one Rh allele from each parent.

Now, let's analyze the given information:

• Mother: $A+$ - Possible genotypes: $I^A I^A$ or $I^A i$ and Rh positive (can be $D D$ or $D d$)
• Father: $B+$ - Possible genotypes: $I^B I^B$ or $I^B i$ and Rh positive (can be $D D$ or $D d$)
• Child: $A+$ - Possible genotypes: $I^A I^A$ or $I^A i$ and Rh positive (can be $D D$ or $D d$)

Now let's see why options B and E are correct:

Option B:

$$I^A I^A\left|I^B i\right| I^A i$$

• Mother: $I^A I^A$ (Blood group A) and Rh positive (can be $D D$ or $D d$)
• Father: $I^B i$ (Blood group B) and Rh positive (can be $D D$ or $D d$)
• Child: $I^A i$ (Blood group A) and Rh positive (can be $D D$ or $D d$)

Option E:

$$I^A i\left|I^B i\right| I^A i$$

• Mother: $I^A i$ (Blood group A) and Rh positive (can be $D D$ or $D d$)
• Father: $I^B i$ (Blood group B) and Rh positive (can be $D D$ or $D d$)
• Child: $I^A i$ (Blood group A) and Rh positive (can be $D D$ or $D d$)

Both Option B and E are valid possibilities because they fulfill the given blood group information for the parents and child.

The other options are incorrect because they either don't match the child's blood group or don't allow for the possibility of the father's blood group.