It is known that $4000$ automobile trips are generated in a large residential area from noon to $1$:$00$ P$.$M$.$ on Saturdays for shopping purposes. Four major shopping centers have the following characteristics:

$(1)$ If a logit model is estimated with coefficients of $-0\mathrm{.}543$ for distance $($in miles$)$ and $0\mathrm{.}0165$ for commercial space $($in thousands of ft$2)$ for all shopping centers, how many shopping trips will be made to each of the four shopping centers?

$(2)$ Consider the shopping trip situation described above. Suppose that shopping center $3$ goes out of business and shopping center $2$ is expanded to $450,000$ ft$2$ of commercial space. What would be the new distribution of the $4000$ Saturday afternoon shopping trips?