## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables and the objective function, as well as the constraints.

- Let \(x_1\) be the number of internet service installations.
- Let \(x_2\) be the number of TV service installations.

The objective is to maximize profit, where the profit per internet service installation is $100 and per TV service installation is $120. So, the objective function can be represented as:

\[ \text{Maximize:} \quad 100x_1 + 120x_2 \]

The constraints are based on the available wiring and box installation times:

- Each internet service takes 60 minutes of wiring time, and each TV service takes 50 minutes. The total available wiring time is 7000 minutes. So, the wiring time constraint is:
\[ 60x_1 + 50x_2 \leq 7000 \]

- Each internet service takes 10 minutes of box installation time, and each TV service takes 20 minutes. The total available box installation time is 4000 minutes. So, the box installation time constraint is:
\[ 10x_1 + 20x_2 \leq 4000 \]

Also, \(x_1 \geq 0\) and \(x_2 \geq 0\) because the number of installations cannot be negative.

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'internet service installations'), ('x2', 'TV service installations')],
    'objective_function': '100*x1 + 120*x2',
    'constraints': [
        '60*x1 + 50*x2 <= 7000',
        '10*x1 + 20*x2 <= 4000',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define the variables
x1 = model.addVar(name="internet_service", lb=0, vtype=gp.GRB.INTEGER)  # Number of internet services
x2 = model.addVar(name="tv_service", lb=0, vtype=gp.GRB.INTEGER)  # Number of TV services

# Define the objective function
model.setObjective(100*x1 + 120*x2, gp.GRB.MAXIMIZE)

# Add constraints
model.addConstr(60*x1 + 50*x2 <= 7000, name="wiring_time_constraint")
model.addConstr(10*x1 + 20*x2 <= 4000, name="box_installation_time_constraint")

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal number of internet service installations: {x1.varValue}")
    print(f"Optimal number of TV service installations: {x2.varValue}")
    print(f"Maximum profit: ${model.objVal:.2f}")
else:
    print("The model is infeasible.")
```