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 $\boxed{{\displaystyle 41164200}}$.