A catering service offers 11 appetizers, 12 main courses, and 10 desserts. A customer is to select 7 appetizers, 8 main courses, and 5 desserts for a banquet. In how many ways can this be done?

Step 1 ：This problem involves choosing a certain number of items from a larger set, where the order in which we choose the items does not matter. This is a combination problem. The formula for combinations is \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.

Step 2 ：We need to calculate the number of combinations for each course separately. For appetizers, we have 11 options and we need to choose 7. Using the combination formula, we get \(C(11, 7) = 330\).

Step 3 ：For main courses, we have 12 options and we need to choose 8. Using the combination formula, we get \(C(12, 8) = 495\).

Step 4 ：For desserts, we have 10 options and we need to choose 5. Using the combination formula, we get \(C(10, 5) = 252\).

Step 5 ：To get the total number of ways to select the menu, we multiply the number of combinations for each course together. So, \(330 \times 495 \times 252 = 41164200\).

Step 6 ：Final Answer: The total number of ways to select the menu is \(\boxed{41164200}\).