You do not have to worry about permutation (counting 2 or more possibilities multiple times) as the positions are unique.
So you can think of these kind of questions step by step.
1st step:
How many ways can you fill the 1st position? (cook)
8 candidates, this 8 ways
2nd step:
For 8 different cooks how many food servers can be selected?
As a persion cannot be cook and server for any cook selected there are 7 different people who can be servers, so this is 8*7 ways
3rd step:
For all the 8*7 ways of cooks there are 6 people who can be cashiers.
So the solution is: 8*7*6=336
