Answer:
Man's genotype: Bbdd
Woman's genotype: bbDd
First child's genotype: Bbdd
Second child's genotype: bbDd
Explanation:
This is a dihybrid cross involving two genes; one coding for eye color and the other for hair color in humans. The allele for brown eye (B) is dominant over the allele for blue eyes (b) in the first gene while the allele for dark hair (D) is dominant over the allele for red hair (d) in the second gene.
According to the question, A man with brown eyes and red hair will possibly possess genotypes: BBdd or Bbdd while a woman with blue eyes and dark hair will possibly have genotype: bbDD or bbDd. Considering the fact that they produced children with recessive traits for both gene (blue eyes and red hair), it means that they are heterozygous for their dominant trait. This means that the ideal genotype for the man is Bbdd since he will produce gametes containing B and b alleles for the first gene while the ideal genotype for the woman is bbDd since it will produce gametes with D and d allele for the second gene.
Hence, a child with brown eyes and red hair will possess genotype: Bbdd since he/she cannot receive two dominant alleles for the first gene from both parents. A child with blue eyes and dark hair will have genotype: bbDd since he/she cannot receive two dominant alleles of the second gene from both parents.
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