If a quantitative character is influenced by the additive effect of four genes how many phenotypic categories are expected in the progeny of a tetrahybrid cross

The correct option is B $9$
For quantitative inheritance by four genes, we have a total of 8 alleles that will determine the phenotypes of the individual. Those offspring with all 8 dominant alleles ( Let's assume it as AABBCCDD ) will show the highest additive effect. Gradually as we keep decreasing the number of dominant alleles, additive effect will become less prominent. Hence, offspring with 7,6,5,4,3,2 and 1 dominant alleles will show different phenotypes. Finally, those individuals with no dominant alleles i.e. all recessive ( aabbccdd ) will show the least additive effect of characters. If we count all of them we have a total of 09 types of phenotypes possible.

So, the correct option is '9'.