How many ways can a cook arrange 9 of her 12 spice jars on a spice rack?

It depends if the 12 spice jars are all different.

If they are:

_ _ _ _ _ _ _ _ _ (9 spaces)

12*11*10*9*8*7*6*5*4

12 possibilities for first one, 11 for second, etc...

Multiply out to get:
79833600

How many jars can be in the first place? 
12
In the second place? 
11 (as one is already in the first place)
Third?
10 
and so on. 
For each put in the first place you can choose 11 in the second and 10 in the third and so on. 
Thus the solution is
12*11*10*9*8*7*6*5*4
or a simpler way to write
12!/3!