A certain car dealership offers its newest model in ten

I don't think so. What's your logic behind 10! and 8!?

Bunuel,
I solved it by 10C2 * 2!(as order is not required)..is that correct?

Yes. 10C2 gives the number of different 2-color combinations possible out of 10 and multiplying by 2 takes into the account that interior can be of one color and the exterior of another or vise-versa.

The question says there are 10 different colors.
Since interior and exterior are to be painted with different colors, the order is important while counting the color combination.
Therefore, the number of possible ways can be found out by permutation of 2 colors out of 10 colors.
10P2 = 10!/(10-2)! = 10!/8! = 90 [10P2 = 10C2*2!]

Answer: (C)

I lined up the 10 different colours for Exterion and Interior like this:

E: A B C D E F G H I J
I: A B C D E F G H I J

I calculated the number of combinations: 10*10= 100 combinations.
But, from these 100 we need to delete sth: the same colours.

There are 10 possible pairs of coulours that will be the same. So, I deleted these 10 pairs from 100, resulting in 90.
Mind here, that the 10 do not result in 20 colours, because 1 pair is one unit that is not allowed. So, 10 units are not allowed (surprised that 80 wasn't among the answer options to confuse test takers...).

One question: I was thinking of the answer given by vince44. Why is it not possible to solve it by 10!/8!.
Could this mean: out of the 10 you deduct the same pairs? 8 will be different, but 2 (one pair) will be the same. So, we deduct this one. Because the 10 are pairs of different colours and 2 are one pair of same colours. So, by the method of anagramming you would have ABCDEFGHSS, where S is the Same colours; which would be 10!/8!...?

Really not sure though. But, reading this solution it had me thinking...

Or would this then be 10!/2!, which is in the answer choices, but is wrong?

Hmmm..

Let's consider the exterior first.

I have 10 options. So if I need to select a colour, it will be 10C1 = 10
Now I have selected 1 colour and that colour cannot be selected for the interior. That means I have 9 interior colours to choose from. That is 9C1 = 9

Now, since it is exterior and interior, we have to multiply. So, 10C1 * 9C1 = 10*9 = 90.
Answer C