# 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?

• Yes, it does. Assuming that an "order" reflects the individual choices. Thus, you are assuming that $ABCD$ is different then $ABDC$ even though someone else might say that in each case the "order" given to the server is "one each of $A,B,C$ and $D$" I'd say the phrasing of the question was ambiguous, myself.
– lulu
Jan 22, 2022 at 17:20
• Shorthand notation is $~\displaystyle \frac{n!}{(n-k)!}~$ where $~(n,k) = (12,4).~$ Compare this with $~\displaystyle \binom{n}{k} = \frac{n!}{(n-k)! \times k!}~$ which represents the number of ways that $k$ distinct orders could be chosen from $n$ orders, without any regard to which of the $k$ people got which orders. Jan 22, 2022 at 18:14

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.
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!