## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the score by determining the number of long answer questions and multiple choice questions to answer, given certain constraints.

Let's define the variables:
- \(x\): the number of long answer questions to answer
- \(y\): the number of multiple choice questions to answer

The objective function to maximize the score is:
\[ \text{Maximize:} \quad 10x + 2y \]

The constraints are:
1. Total questions: \( x + y \leq 15 \)
2. Minimum long answer questions: \( x \geq 5 \)
3. Minimum multiple choice questions: \( y \geq 7 \)
4. Maximum of either type: \( x \leq 11 \) and \( y \leq 11 \)
5. Non-negativity: \( x \geq 0 \) and \( y \geq 0 \), but since we have minimum requirements, this is implicitly satisfied.

## Gurobi Code Formulation

We will use the Gurobi library in Python to solve this problem.

```python
import gurobi

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

    # Define variables
    x = model.addVar(lb=5, ub=11, name="long_answer_questions")  # At least 5, at most 11
    y = model.addVar(lb=7, ub=11, name="multiple_choice_questions")  # At least 7, at most 11

    # Objective function: Maximize 10x + 2y
    model.setObjective(10 * x + 2 * y, gurobi.GRB.MAXIMIZE)

    # Constraint: x + y <= 15
    model.addConstr(x + y <= 15, name="total_questions")

    # Solve the problem
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        # Get the solution
        long_answer_optimal = x.varValue
        multiple_choice_optimal = y.varValue
        max_score = 10 * long_answer_optimal + 2 * multiple_choice_optimal

        print(f"Optimal long answer questions: {long_answer_optimal}")
        print(f"Optimal multiple choice questions: {multiple_choice_optimal}")
        print(f"Maximum score: {max_score}")
    else:
        print("The problem is infeasible.")

solve_problem()
```

However, to adhere strictly to the format requested and to ensure executable code block:


```python
import gurobi

def solve_problem():
    model = gurobi.Model()

    x = model.addVar(lb=5, ub=11, name="long_answer_questions")
    y = model.addVar(lb=7, ub=11, name="multiple_choice_questions")

    model.setObjective(10 * x + 2 * y, gurobi.GRB.MAXIMIZE)

    model.addConstr(x + y <= 15, name="total_questions")

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        long_answer_optimal = x.varValue
        multiple_choice_optimal = y.varValue
        max_score = 10 * long_answer_optimal + 2 * multiple_choice_optimal

        print(f"Optimal long answer questions: {long_answer_optimal}")
        print(f"Optimal multiple choice questions: {multiple_choice_optimal}")
        print(f"Maximum score: {max_score}")
    else:
        print("The problem is infeasible.")

solve_problem()
```