## Step 1: Define the symbolic variables
Let's define the symbolic variables for the quantities mentioned in the problem:
- $x_1$ represents the number of student version software packages produced.
- $x_2$ represents the number of professional version software packages produced.

## Step 2: Formulate the objective function
The profit per student version is $400, and the profit per professional version is $1500. The objective is to maximize the total profit, which can be represented as:
\[ \text{Maximize:} \quad 400x_1 + 1500x_2 \]

## 3: Define the constraints
1. The marketing department estimates that they can sell at most 350 licenses for both versions combined a month:
\[ x_1 + x_2 \leq 350 \]
2. The cost to produce a student version is $750, and a professional version is $3000. The total cost should not exceed $500,000:
\[ 750x_1 + 3000x_2 \leq 500,000 \]
3. Non-negativity constraints, as the number of software packages cannot be negative:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

## 4: Symbolic Representation
The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'student version software packages'), ('x2', 'professional version software packages')],
    'objective_function': '400*x1 + 1500*x2',
    'constraints': [
        'x1 + x2 <= 350',
        '750*x1 + 3000*x2 <= 500000',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 5: Gurobi Code
Now, let's implement this problem using Gurobi in Python:
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="x1", lb=0, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="x2", lb=0, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER)

    # Define the objective function
    model.setObjective(400*x1 + 1500*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(x1 + x2 <= 350)
    model.addConstr(750*x1 + 3000*x2 <= 500000)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of student version software packages: {x1.varValue}")
        print(f"Number of professional version software packages: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```