Ablue-eyed, left-handed woman marries a brown-eyed, right-handed man who is heterozygous for both of his traits. if blue eyes and left-handedness are recessive, how many different phenotypes are possible in their children?

4 different phenotypes may be produced. If you write this down in a Punnet square you can get the exact ratio and the genotypes along with the phenotypes. 

Let us assign B for brown eyes, b for blue eyes; H for right-handed, h for left-handed. 
The mother's genotype will then be bbhh
The father's genotype will then be BbHh

           bh        bh         bh           bh
BH    BbHh    BbHh    BbHh      BbHh  
Bh     Bbhh    Bbhh     Bbhh      Bbhh
bH     bbHh   bbHh     bbHh      bbHh
bh      bbhh    bbhh     bbhh       bbhh

Now based on that you can see that there are 4 possible genotypic combinations which would express different phenotypic combinations as well. 

BbHh: Brown-eyed, right-handed
Bbhh: Brown-eyed, left-handed
bbHh:  Blue-eyed, right handed
bbhh:   Blue-eyed, left-handed

So again, there would be 4 possible phenotypes. 

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