Question $1.$ As an industrial engineer, you are working on a logistics optimization project for ABC Pharmaceuticals. The company aims to deliver its medicines from the central depot to various pharmacies in the most efficient and cost$-$effective way. The distribution network of the company consists of a total of $11$ nodes, including the central depot and $10$ pharmacies. Since pharmacies $2,4,$ and $7$ are located in nearby regions, the company wants these pharmacies to be visited consecutively. Since pharmacies $5$ and $6$ are located in different city centers, it is important that they are not visited consecutively. Additionally, as pharmacy $10$ does not require the delivery of critical or sensitive medicines, it should be visited only after all other pharmacies have been visited.

There is a medicine distribution network consisting of $1$ depot and $10$ pharmacies, making a total of $11$ nodes. The company wants to deliver its medicines to pharmacies on time and in a costeffective manner. The following constraints must be satisfied:

$-$ Pharmacies $2,4,$ and $7$ must be visited consecutively $($the order does not matter$).$

$-$ Pharmacies $5$ and $6$ must not be visited consecutively.

$-$ Pharmacy $10$ must be visited after all other pharmacies.

a$)$ How many feasible delivery routes are there under these conditions?

b$)$ Write the mathematical model for the related TSP problem $($including the constraints that mentioned above.$)$