## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The goal is to maximize the profit by determining the optimal number of internet and TV services to install, given the constraints on wiring time and box installation time.

Let's define the decision variables:

* $x$: number of internet services to install
* $y$: number of TV services to install

The objective function to maximize is the total profit:

* Profit per internet service: $100
* Profit per TV service: $120
* Total profit: $100x + 120y

The constraints are:

* Wiring time: 60 minutes per internet service, 50 minutes per TV service, and 7000 minutes available
* Box installation time: 10 minutes per internet service, 20 minutes per TV service, and 4000 minutes available

The constraints can be formulated as:

* $60x + 50y \leq 7000$ (wiring time constraint)
* $10x + 20y \leq 4000$ (box installation time constraint)
* $x \geq 0$ and $y \geq 0$ (non-negativity constraints)

## Gurobi Code

```python
import gurobipy as gp

# Create a new model
m = gp.Model("Telecom_Service_Installation")

# Define the decision variables
x = m.addVar(name="internet_services", lb=0, vtype=gp.GRB.INTEGER)
y = m.addVar(name="tv_services", lb=0, vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(100*x + 120*y, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(60*x + 50*y <= 7000, name="wiring_time_constraint")
m.addConstr(10*x + 20*y <= 4000, name="box_installation_time_constraint")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: x = {x.varValue}, y = {y.varValue}")
    print(f"Maximum profit: ${100*x.varValue + 120*y.varValue}")
else:
    print("No optimal solution found")
```