Step 1 :The problem is asking for the total number of different pizzas that can be ordered. This is a combination problem where we multiply the number of choices for each category (crust, cheese, meat topping, and vegetable topping) together to get the total number of combinations.
Step 2 :Given that there are 5 types of crust, 2 types of cheese, 5 meat toppings and 7 vegetable toppings.
Step 3 :We calculate the total number of different pizzas that can be ordered by multiplying the number of choices for each category together. This can be represented as \(5 \times 2 \times 5 \times 7\).
Step 4 :Calculating the above expression gives us a total of 350 different pizzas.
Step 5 :Final Answer: The total number of different pizzas that can be ordered at Mama Joan's is \(\boxed{350}\).