Answer: The genotype of the mare is BBCC.
Explanation-
The genotype of the white stallation is Bbcc and brown mare is BBC_. This indicates that this trait is determined by the allele C and c. White stallation (cc) is a recessive trait and brown mare is a dominant trait (allele is C). This means that even one 'C' allele in the genotype will give brown mare.
As per the question, brown mare can only have brown offspring with the white stallation.
This means that the only possible genotype of the parental brown mare is BBCC, which produces all brown mares when crossed with the given white stallation.