Bessie and the girls have rented $4$ barns on the ocean front for Bovine Beach Week. The landlord is rather eccentric: he has required that the first and last barns must contain an odd number of cows, and the two in between must contain an even number of cows. Additionally, Bessie wants to make sure that none of the barns is unused. If there are $10$ cows in total, and they are all distinct (very unique, in fact), how different possible housing arrangements can Bessie choose from?