A man has type A blood and his wife has type B blood. A physician types the blood of their four children and finds each of the children has a different blood type. What are the genotypes of the parents? How would you explain that all four blood types could be present?

Phenotype is a description of a person's physical characteristics. I tell students that phenotypes are written using descriptive words. A person has "blue eyes" or "brown eyes." The phenotype of the dad in this question is "type A blood," and the phenotype of the mom is "type B blood." What the phenotype does not confirm for us is the genotype. Genotype is the genetic identity of a person. The genotype shows what version of a gene (allele) is present on each of the chromosomes in a homologous pair. Because humans are diploid organisms, each person inherited a blood type gene from each parent; however, the inherited gene might not be the same version.
Let me explain using eye color first. A purebred blue-eyed parent will donate a blue-eyed gene. We can represent this gene with a lowercase "b." A purebred brown-eyed parent will donate a brown-eyed gene, and we will show this with a capital "B." When those two genes come together, the hypothetical child now has a blue-eyed gene and a brown-eyed gene. Because of genetic dominance, this child will phenotypically be brown-eyed. Genotypically, the child will be "Bb." The dominant, brown-eyed gene is phenotypically displayed, and the genotype shows that the child is a carrier for the recessive gene.
Blood type works in a similar fashion, but blood type is also codominant. This is why a person can be type AB blood. They carry the B blood gene and the A blood gene, and both blood types are displayed. Things get a bit more complicated when dealing with a person with A blood or B blood because we don't know if they are purebred (homozygous) type A or B. We would need to know their genotype. A person with type A blood could genotypically be "AA" or "AO." In this illustration, the "O" is the recessive version of the gene. You could write "a" instead, but I prefer writing the "O" for students because it explains how type O blood can genetically occur.
Since the question specifies that all four blood groups are represented among the children, that means each parent must have a heterozygous genotype. Dad is AO, and Mom is BO. If Dad donates the A gene and Mom donates the recessive O, then the child is genotypically AO and phenotypically A. This is how the couple could produce a type B child. Both parents donating the dominant blood type would create a child with AB blood, and both parents donating the recessive blood type would result in a type O child. A Punnett square can be used to effectively illustrate this. I've linked one below.
https://cdn6.bigcommerce.com/s-0cordqo/product_images/uploaded_images/punnett-square.png