The genotypes of husband and wife are $I^A I^B$ and $I^A I^O$. Among the blood groups of their children, how many different genotypes and phenotypes are possible?

The cross is between the genotypes $I^A I^B$ and $I^A I^O$.

Gametes produced by the husband are $I^A$ and $I^B$.

Gametes produced by the wife are $I^A$ and $I^O$.

The possible genotypes of the offspring are obtained by the combinations of these gametes:

$I^A I^A$ (Blood group A)

$I^A I^O$ (Blood group A)

$I^A I^B$ (Blood group AB)

$I^B I^O$ (Blood group B)

There are $4$ different genotypes: $I^A I^A$, $I^A I^O$, $I^A I^B$, and $I^B I^O$.

There are $3$ different phenotypes: Blood group A, Blood group AB, and Blood group B.

Thus, $4$ genotypes and $3$ phenotypes are possible.

Answer: $4$ genotypes and $3$ phenotypes