Solution provided is correct! For a better understanding of why you did what you did, let us visualize the problem.
The waiter must remember who ordered which dish which means that he must dinstinguish between the individuals. This allows us to assign a distinct color (or number) to each individual. Now, what we can observe is that the color (or number) is equivalent to ordering them.
Why is that? If the four friends are colored red, yellow, green and blue then we cannot say that the dishes ordered by (red, yellow, green, blue) are the same as dishes ordered by (yellow, blue, red, green).
For each position of dish corresponding to (red, yellow, green, blue) we have 12 possible orders totalling to $12^4$.
Now, what happens when the waiter need not remember who has ordered what dish i.e., only needs to remember final order instead of individual orders? As a hint, look at the following visualization:
Clearly, (red, yellow, green, blue) = (yellow, blue, red, green). How many orders are extra in the first case?