Yup I believe the seniority determines the locant number. Higher seniority = assign lower locant. So since chlorine has higher seniority than bromine it is on second carbon and bromine is on third carbon. In nomenclature when writing prefix we use alphabetical order.
Hence 3-bromo-2-chlorobutane is correct. The numbers are assigned by seniority and the order in which they are written is alphabetical
We were told do them in numerical order too. e.g. 2-methyl,3-ethylpentane
-ethyl and -methyl have no seniority over each other, so should be written alphabetically and then their numbering is then based on their position in the alphabet. 3-ethyl-2-methylpentane is what you should have put.
Whereas halogens show seniority to each other, which overrules alphabetics in assigning their locant position, but not their order when written.