Counting possible number of food orders I am trying to calculate the total number of possible orders from a group of $4$ friends at a restaurant and there are $12$ different dishes to choose from. Each friends has to have a different dish from each other. So far, I have this calculation:
$12 * 11 * 10 * 9 = 11,880$
Does this look right?
 A: This looks right! Another way to think about this is the first friend orders and has $12$ choices. The second friend can then order anything that the first friend didn't, so they have $11$ choices. The third friend similarly has $10$ choices and the final friend has $9$ choices.
So by the multiplication principle, the total number of orders is $12 \ast 11 \ast 10 \ast 9$ as you stated.
As someone mentioned in a comment, it is worthwhile to note this is different from asking the question: "What are all the possible ways to choose $4$ distinct dishes from the $12$ possible dishes?" Since in your situation, we do care which friend gets which meal.
A: The formal method to calculate this is to use permutation notation. In this case we have $n = 12$ dishes to choose from and $k = 4$ people choosing things. We also have that they cannot chose the same dish. So we use the formula
\begin{align}
     \frac{n!}{(n - k)!} &= \frac{12!}{(12 - 4)!}\\
                         &= \frac{12!}{8!}\\
                         &= \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8!}{8!}\\
                         &= 12 \cdot 11 \cdot 10 \cdot 9 = 11,880
\end{align}
so you are correct!
