SOLUTION: I cannot seem to wrap my head around this one... compute total number of ways in which 3 differently colored bottles could be arranged in a line out of a total of 9 bottles?

We have 9 things and we must position 3 of them.

That's "9 Position 3"

9P3 = 9*8*7 = 504

Or you can look at it this way:

There are 9 ways to choose the bottle to go on the far left.

For each of those 9 ways of placing a bottle on the far left, there are
8 ways to choose a bottle to go in the middle. That's 9*8 or 72.

For each of those 9*8 or 72 ways of placing the first two bottles
on the far left and in the middle, there are 7 ways to choose a bottle 
to go on the far right. That's 9*8*7 or 504.

Edwin